Program and Abstracts

Riemannian Geometry and Related Fields

⋅ 4th-A-09:00 − 10:00   Chair: Young Jin Suh (Kyungpook National University)

⋅ 4th-A-09:00 − 09:30 Generalized vector cross products (Uwe Semmelmann)

Uwe Semmelmann, University of Stuttgart

In my talk I will introduce the notion of generalized vector cross products on $\mathbb R^n$ and describe their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of SU(3)-structures in dimension 6 whose associated 3-form is Killing, I will show that every Killing 3-form on a compact $n$-dimensional Riemannian manifold with negative sectional curvature vanishes if $n\ge 4$. The talk is based on a joint work with Laura Barberis and Andrei Moroianu.

2010 Mathematics Subject Classification: 53C21, 15A69
Key Words and Phrases: Killing tensors, Killing forms, vector cross products, SU(3)-structures

⋅ 4th-A-09:30 − 10:00 Lagrangian geometry of the Gauss images of isoparametric hypersurfaces (Yoshihiro Ohnita)

Yoshihiro Ohnita, Osaka City University Advanced Mathematical Institute

The images of the Gauss map (Gauss images) of isoparametric hypersurfaces in the standard sphere provide a nice class of compact minimal (thus monotone) Lagrangian submanifolds embedded in a complex hyperquadrics which is a compact Hermitian symmetric space of rank 2. It is an interesting problem to study the properties of such Lagrangian submanifolds by the well-developed theory of isoparametric hypersurfaces. In this talk I will give a survey of our recent works and environs on Hamiltonian stability problem, Hamiltonian non-displaceability and Floer homology of such Lagrangian submanifolds. I will also mention about some related open problems and conjectures. This talk is mainly based on my joint works on Hui Ma (Tsinghua University, P. R. China), Hiroshi Iriyeh (Ibaraki University, Japan) and Reiko Miyaoka (Tohoku University, Japan).

2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: isoparametric hypersurfaces, minimal Lagrangian submanifolds, complex hyperquadrics, Gauss images, Lagrangian intersections

⋅ 4th-B-13:30 − 15:30   Chair: Uwe Semmelmann (University of Stuttgart)

⋅ 4th-B-13:30 − 14:00 Maximal antipodal sets of classical compact symmetric spaces (Makiko Sumi Tanaka)

Makiko Sumi Tanaka, Tokyo University of Science

A subset $S$ of a compact Riemannian symmetric space is called an antipodal set if it satisfies $s_x(y)=y$ for any $x, y$ in $S$, where $s_x$ denotes the geodesic symmetry at $x$. An antipodal set is finite. It is known that the intersection of two real forms in a Hermitian symmetric space of compact type is an antipodal set if they intersect transversely. A maximal antipodal set of a symmetric $R$-space is unique up to congruence and it is an orbit of some Weyl group. A maximal antipodal set of a compact Lie group which contains the identity element is a subgroup, which is isomorphic to an abelian group $\mathbb{Z}_2 \times \cdots \times \mathbb{Z}_2$. We classified maximal antipodal subgroups of the quotient groups of some classical compact Lie groups by giving explicit description of representative of each conjugate class. In this talk I will explain the classification of maximal antipodal sets of some compact Riemmanian symmetric spaces of classical type and their quotient spaces. We use nice embeddings of these symmetric spaces into compact Lie groups and our former results. This talk is based on a joint work with Hiroyuki Tasaki.

2010 Mathematics Subject Classification: 53C35
Key Words and Phrases: compact symmetric space, antipodal set, compact Lie group

⋅ 4th-B-14:00 − 14:30 Hopf real hypersurfaces in the indefinite complex projective space (Makoto Kimura, Miguel Ortega)

Makoto Kimura, Ibaraki University, Miguel Ortega*, University of Granada

H.~Anciaux and K.~Panagiotidou obtained the very basic results for real hypersurfaces in the indefinite complex projective space ${\mathbb C}P^n$ of index $1\leq p\leq n-1$. (See H. Anciaux, K. Panagiotidou, \emph{Hopf Hypersurfaces in pseudo-Riemannian complex and para-complex space forms}, Diff. Geom. Appl. \textbf{42} (2015), 1--14). We wish to further develop their ideas. We re-obtain the almost contact metric structure on a real hypersurface in ${\mathbb C}P^n$, namely $(g,\phi,\xi,\eta)$. We construct new examples, which we call \textit{real hypersurfaces of type $A_+$, $A_-$, $B_0$, $B_+$, $B_-$ and $C$}, because they are somehow similar to those in the famous Takagi's and Montiel's lists. All our examples are Hopf. We also exhibit an example of a lightlike Hopf real hypersurface. We prove that two real hypersurfaces with the same shape operator are linked by a holomorphic isometry of the ambient space. Next, we classify the non-degenerate real hypersurfaces in indefinite complex projective space such that $AX=\lambda X+\rho \eta(X)\xi$, where $\lambda$ and $\rho$ are smooth functions. Finally, we classify those non-degenerate real hypersurfaces such that $A\phi=\phi A$.

2010 Mathematics Subject Classification: Primary 53B25, 53C50; Secondary 53C42, 53B30
Key Words and Phrases: real hypersurface, indefinite complex projective space, Hopf real hypersurface

⋅ 4th-B-14:30 − 15:00 Lie invariant and Ricci tensor on real hypersurfaces in complex two-plane Grassmannians (Chang Hwa Woo, Imsoon Jeong, Young Jin Suh, Juan de dios Perez)

우창화*((우석대)), 정임순((배재대)), 서영진((경북대)), Juan de dios Perez((University of Gra\-nada))
Chang Hwa Woo*, Woosuk University, Imsoon Jeong, Paichai University, Young Jin Suh, Kyungpook National University, Juan de dios Perez, University ofGranada

On a real hypersurface $M$ in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ we have Lie derivative and a differential operator of order one associated to the generalized Tanaka-Webster connection $\widehat {\mathcal L} ^{(k)}$. We give a classification of real hypersurfaces $M$ on $G_2({\mathbb C}^{m+2})$ satisfying $\widehat {\mathcal L} ^{(k)}_{\xi}S={\mathcal L}_{\xi}S$, where $\xi$ is the Reeb vector field on $M$ and $S$ the Ricci tensor of $M$.

2010 Mathematics Subject Classification: 53C40, 53C15
Key Words and Phrases: real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurface, shape operator, Ricci tensor, Lie derivation

⋅ 4th-B-15:00 − 15:30 Orbifolds with all geodesics closed (Manuel Amann, Christian Lange, Marco Radeschi)

Manuel Amann*, University of Augsburg, Christian Lange, University of Cologne, Marco Radeschi, University of Notre Dame

The concept of a Riemannian orbifold generalises the one of a Riemannian manifold by permitting certain singularities. In particular, one is able to speak about several concepts known from classical Riemannian geometry including geodesics. Whenever all geodesics can be extended for infinite time and are all periodic, the orbifold is called a \emph{Besse orbifold}---in analogy to Besse manifolds. A classical result in the simply-connected manifold case states that in odd dimensions only spheres may arise as examples of Besse manifolds. In this talk we shall illustrate that the same holds for Besse orbifolds, namely that they are actually already manifolds whence they are spheres. The talk is based on joint work in progress with Christian Lange and Marco Radeschi.

2010 Mathematics Subject Classification: 57N65
Key Words and Phrases: orbifold, geodesic, Besse, equivariant cohomology, loop space

⋅ 5th-E-13:30 − 15:30   Chair: Yoshihiro Ohnita (Osaka University Advanced Mathematical Institute)

⋅ 5th-E-13:30 − 14:00 Antipodal sets of generalized $s$-manifolds (Shinji Ohno, Takashi Sakai, Yasunori Terauchi)

Shinji Ohno, Nihon University, Takashi Sakai*, Tokyo Metropolitan University, Yasunori Terauchi, Tokyo Metropolitan University

In this talk, we give the definition of generalized $s$-manifolds as a generalization of symmetric spaces. For a generalized $s$-manifolds, we introduce the notions of polar and antipodal set, and define a geometric invariant, which we call the antipodal number, as the supremum of cardinalities of antipodal sets. We show that a flag manifold consisting of sequences of subspaces with inclusion relations in a vector space admits structures of generalized $s$-manifolds, and determine maximal antipodal sets and the antipodal number. We mention that the antipodal number of the flag manifold is related to its topology.

2010 Mathematics Subject Classification: 53C35
Key Words and Phrases: antipodal set, symmetric space, flag manifold

⋅ 5th-E-14:00 − 14:30 Ricci soliton and pseudo-Einstein real hypersurfaces in the complex hyperbolic quadric (Young Jin Suh, Hyunjin Lee, Gyu Jong Kim)

서영진*((경북대)), 이현진((경북대)), 김규종((우석대))
Young Jin Suh*, Kyungpook National University, Hyunjin Lee, Kyungpook National University, Gyu Jong Kim, Woosuk University

First we introduce a new notion of Ricci-soliton and pseudo-Einstein for real hypersurfaces in the noncompact complex hyperbolic quadric $SO^0_{2,m}/SO_2 SO_m$. Next as an application we will prove non-existence properties of Ricci soliton and pseudo-Einstein real hypersurfaces in the complex hyperbolic quadric $SO^0_{2,m}/SO_2 SO_m$.

2010 Mathematics Subject Classification: 53C40, 53C55
Key Words and Phrases: Ricci soliton, pseudo-Einstein, pseudo-anti commuting Ricci tensor, $\mathfrak A$-isotropic, $\mathfrak A$-principal, complex conjugation, complex hyperbolic quadric

⋅ 5th-E-14:30 − 15:00 Derivatives of a certain operator on a real hypersurface in non-flat complex space forms (Juan de Dios Perez)

Juan de Dios Perez, University of Granada

On a real hypersurface $M$ in a non-flat complex space form we can consider two types of connections: the Levi-Civita connection, which is a torsion free connection, and, for any nonnull constant $k$, the $k$-th generalized Tanaka-Webter connection, which is affine connection with torsion. Therefore, we have the associated covariant derivatives. Moreover on $M$ we also have the Lie derivative and a derivative of Lie type associated to the $k$-th generalized Tanaka-Webster connection. We classify real hypersurfaces in non-flat complex space forms for which either both covariant derivatives or Lie derivative and the derivative of Lie type associated to the $k$-th generalized Tanaka-Webster connection coincide when they act on the operator $\phi A-A\phi$, where $\phi$ denotes the structure operator and $A$ the shape operator of $M$, either in the direction of the structure vector field $\xi$ or in any direction of the maximal holomorphic distribution on $M$.

2010 Mathematics Subject Classification: 53C15
Key Words and Phrases: non-flat complex space form, real hypersurface, $k$-th generalized Tanaka-Webster connection

⋅ 5th-E-15:00 − 15:30 On homogeneous manifolds whose isotropy actions are polar (Jos\'e Carlos D\'iaz Ramos, Miguel Dom\'inguez V\'azquez, Andreas Kollross)

Jos\'e Carlos D\'iaz Ramos, University of Santiago de Compostela, Miguel Dom\'inguez V\'azquez, Institute of Mathematical Sciences (CSIC-UAM-UC3M-UCM), Spain, Andreas Kollross*, University of Stuttgart

Symmetric spaces are a central class of examples in Riemannian geometry and it has often been a fruitful approach to study to what extent certain properties of symmetric spaces continue to hold for more general classes of Riemannian manifolds. In this talk, I will speak about the question whether a well-known property of Riemannian symmetric spaces, namely the polarity of their isotropy actions, holds for more general types of homogeneous manifolds. I will present various results on homogeneous spaces with polar isotropy.

2010 Mathematics Subject Classification: 53C30, 53C35, 57S15, 57S20
Key Words and Phrases: homogeneous space, isotropy action, isotropy representation, polar action, Riemannian symmetric space, generalized Heisenberg group, Damek-Ricci space
Complex geometry

⋅ 4th-A-09:00 − 10:00   Chair: Jun-Muk Hwang (KIAS)

⋅ 4th-A-09:00 − 09:30 A product formula for volumes of divisors via Okounkov bodies (Jinhyung Park)

박진형((고등과학원))
Jinhyung Park, KIAS

In this talk, I discuss a generalization of Kawamata's product formula for volumes of canonical divisors to arbitrary divisors using Okounkov bodies. This is joint work with Sung Rak Choi, Seung-Jo Jung, and Joonyeong Won.

2010 Mathematics Subject Classification: 14C20
Key Words and Phrases: Okounkov body, divisor, volume

⋅ 4th-A-09:30 − 10:00 Deformations of hyperk\"ahler twistor spaces (Ana-Maria Brecan, Tim Kirschner, Martin Schwald)

Ana-Maria Brecan, University of Bayreuth, Tim Kirschner*, University of Duisburg-Essen, Martin Schwald, University of Duisburg-Essen

We obtain novel results concerning the deformation theory of twistor spaces of irreducible holomorphic symplectic manifolds. Specifically we show that the local deformations of such twistor spaces are unobstructed. The proof is based upon an extension theorem for families of irreducible holomorphic symplectic manifolds. This theorem compensates for the fact that, generally, fine moduli spaces of marked irreducible holomorphic symplectic manifolds do not exist.

2010 Mathematics Subject Classification: 32G08, 32L25
Key Words and Phrases: unobstructed deformation, twistor space, irreducible holomorphic symplectic manifold

⋅ 4th-B-13:30 − 15:30   Chair: Thomas Peternell (University of Bayreuth)

⋅ 4th-B-13:30 − 14:10 Moduli spaces of cubic threefolds — Geometry and Topology (S. Casalaina-Martin, S. Grushevsky, Klaus Hulek, R. Laza)

S. Casalaina-Martin, University of Colorado, S. Grushevsky, Leibniz University of Hanover, Klaus Hulek*, Leibniz University of Hanover, R. Laza, Stony Brook University

Cubic threefolds were the first class of varieties who were shown to be unirational but not rational (Clemens, Griffiths). The key tool of the proof is the intermediate Jacobian, a principally polarized abelian variety of dimension 5. There is a second link to Hodge theory, namely via cubic fourfolds (Allcock, Carlson, Toledo) which leads to a $10$-dimensional ball quotient model. Looking at cubic threefolds from these different points of view leads to various geometrically relevant compactifications of the moduli space of cubic threefolds. In this talk I will discuss the geometry and the topology of these spaces. This is joint work with S. Casalaina-Martin, S. Grushevsky and R. Laza.

2010 Mathematics Subject Classification: 14J10, 14J70
Key Words and Phrases: cubic threefolds, intermediate Jacobians, moduli spaces

⋅ 4th-B-14:10 − 14:50 Local numerical equivalence via Okounkov bodies (Sung Rak Choi, Jinhyung Park, Joonyeong Won)

최성락*((연세대)), 박진형((고등과학원)), 원준녕((기초과학연구원))
Sung Rak Choi*, Yonsei University, Jinhyung Park, KIAS, Joonyeong Won, IBS-CGP

We study what kind of local numerical properties is encoded in the Okounkov bodies. More precisely, we show that the set of Okounkov bodies of a pseudoeffective divisor with respect to admissible flags centered at a fixed point determines the local numerical equivalence class of divisors which is defined in terms of refined divisorial Zariski decompositions. Our results extend Ro\'{e}'s previous work on surfaces to higher dimensional varieties

2010 Mathematics Subject Classification: 14C20
Key Words and Phrases: Okounkov body, Zariski decomposition, local numerical equivalence

⋅ 4th-B-14:50 − 15:30 Singular varieties with numerically trivial canonical divisor (Andreas H\"oring, Thomas Peternell)

Andreas H\"oring*, University of Nice Sophia Antipolis, Thomas Peternell, University of Bayreuth

The Beauville-Bogomolov decomposition of smooth projective manifolds is a cornerstone of the classification of higher-dimensional varieties. For varieties with mild singularities (e.g. minimal models) the analytic tools used in the proof are not available, so one has to look for a different strategy. In this talk I will explain how foliation theory enters the picture. A positivity result for reflexive sheaves then allows to extend the decomposition theorem to the singular setting. This is joint work with Thomas Peternell.

2010 Mathematics Subject Classification: 14J32, 37F75, 14E30
Key Words and Phrases: MMP, minimal models, algebraic integrability, positivity of vector bundles

⋅ 5th-E-13:30 − 15:30   Chair: Klaus Hulek (Leibniz University of Hanover)

⋅ 5th-E-13:30 − 14:10 On the B-semiampleness conjecture (Enrica Floris, Vladimir Lazi\'c)

Enrica Floris, University of Saarland, Vladimir Lazi\'c*, University of Saarbrucken

The B-Semiampleness Conjecture of Prokhorov and Shokurov predicts that the moduli part in a canonical bundle formula is semiample on a birational modification. We prove that the restriction of the moduli part to any sufficiently high divisorial valuation is semiample, assuming the conjecture in lower dimensions. This is joint work with Enrica Floris.

2010 Mathematics Subject Classification: 14N30, 14E30
Key Words and Phrases: canonical bundle formula, B-Semiampleness conjecture

⋅ 5th-E-14:10 − 14:50 Automorphism groups of the complements of hypersurfaces (Jihun Park)

박지훈((포항공대))
Jihun Park, POSTECH

It is a well-known fact that every automorphism of a smooth hypersurface $S$ of degree $d$ in $\mathbb{P}^n$, $n\geq 2$, comes from the automorphism group of $\mathbb{P}^n$ unless $(n,d)= (2,3), (3,4)$. In my talk, I reinvestigate this phenomenon inside out, i.e., the problem when the automorphism group of the complement of the hypersurface in $\mathbb{P}^n$ coincides with the subgroup of the automorphism group of $\mathbb{P}^n$ that keeps the hypersurface $S$ fixed.

2010 Mathematics Subject Classification: 14E07, 14J45, 14J70, 14R20
Key Words and Phrases: automorphism, birational automorphism, hypersurface, cylinder, unipotent group action

⋅ 5th-E-14:50 − 15:30 Invariants from a generalized polarizations (Christian Lehn)

Christian Lehn, Chemnitz University of Technology

When considering the action of a finite group $G$ on a complex vector space $V$, polarization is an important concept in invariant theory. It allows to produce polynomial invariants of $V^n \oplus (V^*)^m$ out of polynomial invariants of $V$. We discuss this question for $n=1=m$. This is joint work in progress with M. Bulois, M. Lehn, and R. Terpereau.

2010 Mathematics Subject Classification: 13A50, 14L24, 17B63, 14L30
Key Words and Phrases: invariant theory, finite groups, Poisson algebras
Algebraic surfaces, their moduli and related topics

⋅ 5th-D-09:00 − 10:00   Chair: Yongnam Lee (KAIST)

⋅ 5th-D-09:00 − 09:30 On the degree of the canonical map of a surface isogenous to a product (Christian Gleissner)

Christian Gleissner, University of Bayreuth

In this talk we study the degree the canonical map of surfaces isogenous to a product. According to A. Beauville, the degree of the canonical map of a surface $S$ of general type is at most 36. The maximum can only be achieved if $S$ is a ball quotient, $|K_S|$ is base point free, $p_g(S)=3$ and $q(S)=0$. We prove that the highest degree of a regular surface $S$ isogenous to a product, that is obtained by the action of an abelian group is $32$. This value is realized by precisely two families with $p_g(S)=3$ and $q(S)=0$. Moreover we give an example of a surface $S$ isogenous to a product with $p_g(S)=3, q(S)=1$ such that the degree of the canonical map is $24$. This is joint work with Carlos Rito and Roberto Pignatelli.

2010 Mathematics Subject Classification: 14J10, 14L30
Key Words and Phrases: surfaces of general type, group actions, abelian covers

⋅ 5th-D-09:30 − 10:00 Intersection cohomology of the moduli space of sheaves on local CY $3$-fold via Kirwan's desingularization (Kiryong Chung, Youngho Yoon)

정기룡*((경북대)), 윤영호((서울대))
Kiryong Chung*, Kyungpook National University, Youngho Yoon, Seoul National University

Let $M^\circ$ be the space of twisted ideal sheaves $\mathcal{I}_{L,Q}(1)$ where $Q$ is a rank $4$ quardric hypersurface in $\mathbb{P}^r$ and $L$ is a linear subspace of dimension $r-2$. The Simpson compactification $\mathbf{M}$ of $M^\circ$ has been studied in different view points: The homological mirror symmetry and the log minimal model program. In the second perspective, the first named author (in other work with H.-B. Moon) proved that Kirwan's partial desingularization of $\mathbf{M}$ is isomorphic to the moduli space of degree $2$ stable maps in Grassmannian $\mathrm{Gr}(r-1,r+1)$. In this talk, by applying Kirwan's method, we obtain the intersection Hodge-Deligne (IHD) polynomial of $\mathbf{M}$. As a direct consequence, we obtain the IHD-polynomial of the moduli space of pure sheaves on the total space of the canonical line bundle of del-Pezzo surfaces. We also calculate the IHD-polynomial of some open cones which naturally comes from the microlocal geometric structure of $\mathbf{M}$.

2010 Mathematics Subject Classification: 14B05, 14F43, 14N35
Key Words and Phrases: partial desingularization, Hodge-Deligne polynomial, open cone of a link

⋅ 6th-G-14:00 − 15:00   Chair: Insong Choe (Konkuk University)

⋅ 6th-G-14:00 − 14:30 Double Kodaire fibrations and automorphisms of curves (Michael L\"onne)

Michael L\"onne, University of Bayreuth

We review a construction of Kodaira fibrations, i.e., surfaces of general type with a non-isotrivial fibration, which are differentiable fibre bundles. It uses automorphisms of curve which are \emph{disjoint} in the sense that their graphs do not intersect. We discuss certain natural monodromy maps and how they determine the signature and other topological invariants. In particular we will present results obtained in joint work with Ju A Lee and S\"onke Rollenske.

2010 Mathematics Subject Classification: 14J29, 14D05, 57R22
Key Words and Phrases: Kodaira fibration, automorphisms of algebraic curves

⋅ 6th-G-14:30 − 15:00 Rational homology balls and Mori sequences of antiflips (Heesang Park, Dongsoo Shin, Giancarlo Urz\'ua)

박희상((건국대)), 신동수*((충남대)), Giancarlo Urz\'ua ((Pontifical Catholic University of Chile))
Heesang Park, Konkuk University, Dongsoo Shin*, Chungnam National University, Giancarlo Urz\'ua, Pontifical Catholic University of Chile

We detect rational homology balls embedded in smooth $4$-manifolds via the minimal model program for $3$-dimensional complex algebraic varieties. We show that there are (infinitely many) pairs of disjoint rational homology balls $B_{p,q}$ smoothly embedded (i) in regular neighborhoods of (almost all) linear chains of $2$-spheres with negative self-intersections, (ii) in blown-up rational homology balls $B_{n,a} \sharp \overline{\mathbb{CP}^2}$, and (iii) in the Milnor fibers of certain cyclic quotient surface singularities. Furthermore, we show that every embedding of a rational homology in a regular neighborhood of negative $2$-spheres can be derived from the minimal model program if the corresponding rational blow-up of the rational homology ball may be obtained from the regular neighborhood just by a sequence of ordinary blow-ups. This is a joint work with Heesang Park and Giancarlo Urzúa.

2010 Mathematics Subject Classification: 57R40, 57R55, 14B07
Key Words and Phrases: antiflip, Mori sequence, rational homology ball

⋅ 6th-G-15:00 − 16:30   Chair: Seonja Kim (Chungwoon University)

⋅ 6th-G-15:00 − 15:30 Supersingular quartic surfaces (Junmyeong Jang)

장준명((울산대))
Junmyeong Jang, University of Ulsan

A K3 surface defienc over a field of positive characteristic is supersingular if the height of the formal Brauer group is the infinite. This condition is equivalent to that the rank of the Neron-Severi group of a K3 surface is 22. When the base characteristic is not equal to 3, it is also equivalent to that a K3 surface is unirational. In this talk, we will see that, when $k$ is an algebraically closed field of odd characteristic $p$ (possibly except $p=5$), every supersingular K3 surface over $k$ is isomorphic to a smooth quartic surface in $\mathbb{P}_{k}^{P3}$.

2010 Mathematics Subject Classification: 14J28, 14G17
Key Words and Phrases: supersingular K3 surfaces, quartic surfaces

⋅ 6th-H-16:00 − 16:30 Moduli spaces of rank two vector bundles and Hilbert schemes of surface scrolls (Youngook Choi, Flaminio Flamini, Seonja Kim)

최영욱*((영남대)), Flaminio Flamini((University of Rome Tor Vergata)), 김선자((청운대))
Youngook Choi*, Yeungnam University, Flaminio Flamini, University of Rome Tor Vergata, Seonja Kim, Chungwoon University

In this talk, we discuss the Brill Noether locus of rank 2, stable vector bundles of suitable degrees with at least $d-2g+4$ sections on a general $k$-gonal curve. We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which turn out to be special and stable.

2010 Mathematics Subject Classification: 14H60, 14D20, 14J26
Key Words and Phrases: stable rank 2 bundles, Brill-Noether loci, general $k$-gonal curves, Hilbert schemes

⋅ 6th-H-16:30 − 18:00   Chair: Young-Hoon Kiem (Seoul National University)

⋅ 6th-H-16:30 − 17:00 Rigid varieties and the solution of a problem of Morrow Kodaira (Ingrid Bauer)

Ingrid Bauer, University of Bayreuth

In the book ``Complex Manifolds" by J. Morrow and K. Kodaira the following problem was posed: \noindent PROBLEM: Does there exist a rigid compact complex manifold, which is not infinitesimally rigid. We will recall different notions of rigidity and some newly obtained results concerning rigid complex surfaces. Then we will give for each dimension $d \geq 2$ examples of rigid but not infinitesimally rigid varieties.

2010 Mathematics Subject Classification: 14B12, 14L30, 14J10, 14J29, 14E20, 32G05
Key Words and Phrases: rigid complex manifolds, surfaces of general type, deformation theory

⋅ 6th-H-17:00 − 17:30 On cascades of toric log del Pezzo surfaces (DongSeon Hwang)

황동선((아주대))
DongSeon Hwang, Ajou University

There have been numerous attempts to classify log del Pezzo surfaces. To achieve this goal, we proposed an approach by using the concept of a `cascade', which can be viewed as a generalization of the classification of the smooth del Pezzo surfaces. So far this approach has been reasonably successful in the case when the log del Pezzo surface is toric or is of Picard number one. In this talk, I will focus on the toric case and explain the concept of cascades and how it leads to the classification, and possible applications.

2010 Mathematics Subject Classification: 14M25, 14J26, 52B20
Key Words and Phrases: toric surface, log del Pezzo surface, cascades

⋅ 6th-H-17:30 − 18:00 The double point formula with isolated singularities, and canonical embeddings (Fabrizio Catanese)

Fabrizio Catanese, University of Bayreuth

Severi established in 1902 the double point formula for the number of improper double points of a variety X of dimension equal to the codimension. This was done in terms of projective invariants, called ceti. The modern version is given (in Hartshorne's book without reference to Severi) in terms of the Chern classes of the tangent bundle of the normalization of X and of the hyperplane divisor. The main motivation for looking for generalizations is the following theorem, obtained with Keiji Oguiso (the result when X is smooth was obtained in my AMS 1995 Santa Cruz article): If a surface has geometric genus 5 and its canonical map is an embedding of the canonical model X, then X is a complete intersection of type (2,4) or (3,3). After explaining the classical and modern versions of the double point formulae, as given by Fulton and Laksov, and the relation between them, I shall present several generalizations, some over the complex numbers, some char free. I shall moreover briefly overview recent results concerning canonical surfaces in projective spaces of dimension 4 and 5. I shall then illustrate more general embedding obstructions derived from the double point formula, and shall deal with the problem of canonical embeddings of ample hypersurfaces in Abelian varieties.

2010 Mathematics Subject Classification: 14J29, 14J10, 14M07
Key Words and Phrases: Chern classes, improper double points, Gauss map, rational double points
Logic and Computing

⋅ 4th-A-09:00 − 10:00   Chair: Sunyoung Kim (Ewha Womans University)

⋅ 4th-A-09:00 − 09:30 Geometric stability theory for $\mu$-structures (Jung Wook Lee)

이정욱((University of Wrocław))
Jung Wook Lee, University of Wrocław

We introduce a notion of $\mu$-structures which are certain locally compact group actions and prove some counterparts of results on Polish structures introduced by Krupinski. Using the Haar measure of locally compact groups, we introduce an independence, called $\mu$-independence, in $\mu$-structures having good properties. With this independence notion, we develop geometric stability theory for $\mu$-structures. Then we see some structural theorems for compact groups which are $\mu$-structure. We also give examples of profinite structures where $\mu$-independence is different from $nm$-independence introduced by Krupinski for Polish structures.

2010 Mathematics Subject Classification: 03C45, 28E15, 20E08
Key Words and Phrases: mu-structure, mu-independence, Haar measure, geometric stability

⋅ 4th-A-09:30 − 10:00 Mekler's construction on NTP$_1$ theory (JinHoo Ahn)

안진후((연세대))
JinHoo Ahn, Yonsei University

Mekler has developed a way of interpreting any structure as a pure group. These groups are nilpotent of class 2 and of exponent $p>2$. It is known that this construction preserves various model-theoretic properties, such as stability, simplicity, and NTP$_2$ so that it gives us certain pure group examples. In this talk, I will introduce the Mekler construction, and briefly show the main result that this construction also preserves NTP$_1$.

2010 Mathematics Subject Classification: 03C45, 03C98
Key Words and Phrases: Mekler's construction, groups, tree indiscernibility, NTP1 theories

⋅ 4th-B-13:30 − 15:30   Chair: Martin Ziegler (KAIST)

⋅ 4th-B-13:30 − 14:00 First-order theory of internally categorical second-order arithmetic (Stella Moon)

Stella Moon, University of California-Irvine

Wilkie's theorem tells us that Peano arithmetic (PA) is the minimal first-order theory captured by the schematisation of the restricted second-order categorical axiomatisation of the standard model $\mathbb{N}$ over some finite first-order theory -- that is, the first-order schematisation Scheme$(\Phi)$ of the second order axiom $\forall X \Phi(X)$ is such that Scheme$(\Phi) \vdash_{FOL} PA$. But it fails to provide us with a maximal first-order theory, which suggests that external categoricity of arithmetic is not enough to present us with the corresponding unique first-order theory. The situation is different when internal categoricity is involved -- we say that a second-order theory $T^2$ is internally categorical if, in second-order logic with arithmetical language, it is deducible that all models of $T^2$ are isomorphic to one another -- that is $$\vdash_{SOL} \big(\forall M^2, N^2 \models T^2\ (M^2 \cong N^2) \big).$$ We will prove that the first-order theory captured by an internally categorical theory is exactly the theorems of second-order PA in the first-order language of arithmetic.

2010 Mathematics Subject Classification: 03C85, 03B15, 03B30, 03C62
Key Words and Phrases: internal categoricity, models of Peano arithmetic, second order logic, Henkin, full

⋅ 4th-B-14:00 − 14:30 Model checking on sparse structures and beyond (Kord Eickmeyer)

Kord Eickmeyer, Technical University of Darmstadt

While model checking, i.e., the algorithmic problem of checking whether a given formula is true in a given finite structure, is easily seen to be PSPACE complete in most interesting settings and remains hard when parameterized by the input formula, efficient algorithms have been found for restricted classes of structures such as planar graphs. Since model checking provides a unified approach to various algorithmic problems, such results are often referred to as algorithmic meta-theorems. We will review known algorithmic meta-theorems, which have focussed on /sparse/ structure, in a sense that has only recently been made precise. We will then see why current techniques are unlikely to produce results for non-sparse structures and investigate possible techniques for breaking this barrier.

2010 Mathematics Subject Classification: 03C13, 05C85, 68R10
Key Words and Phrases: model checking, sparse graphs

⋅ 4th-B-14:30 − 15:00 Aspects of logic and computing in real number models (Klaus Meer)

Klaus Meer, Brandenburg University of Technology

We present an extract of own work over the past years focussing on logical aspects in the study of real number computation and complexity as introduced by Blum, Shub, and Smale. After a brief introduction into the model we address the following topics: Structure inside the real number analogue of NP, degrees of undecidability, and descriptive complexity over the reals.

2010 Mathematics Subject Classification: 68Q05, 68Q15, 68Q19
Key Words and Phrases: real number computations, complexity classes

⋅ 4th-B-15:00 − 15:30 A Possible combination of concepts from set theory and Le\'sniewski's Mereology for an alternative to ZFC (Jin Hoo Lee)

이진후((한국과학기술원))
Jin Hoo Lee, KAIST

It is well-known that ZFC has long been a firm foundation for modern mathematics. However, due to its limits such as the undecidability of the continuum hypothesis, several mathematicians and logicians have thought up possible supplementations for ZFC or have created alternative foundations (such as NBG, Leśniewski's mereology, etc.) to it. In this talk I will present my trial of combining concepts from set theory, type theory, and Leśniewski's mereology to suggest a possible alternative to ZFC and show how mathematical concepts such as binary relations, functions, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ can be defined. Although my trial contained contradictions and thus failed, I would like to emphasize that adding the concept of parthood from mereology could be a possible supplementation for current set theories.

2010 Mathematics Subject Classification: 03A99
Key Words and Phrases: foundation, parthood, mereology, set theory, alternative set theory, le\'sniewski

⋅ 5th-E-13:30 − 15:30   Chair: Gyesik Lee (Hankyong University)

⋅ 5th-E-13:30 − 14:00 First order G\"odel Logics with propositional quantifiers (Norbert Preining)

Norbert Preining, Accelia Inc., Japan

First order G\"odel logics form a well established class of many-valued logics with good proof-theoretic properties and extensive theory. Quantified Propositional G\"odel Logics have been studied only a few times, and not much is known about these logics besides a few results for specific logics. Propositional quantifiers allow for quantification of propositions, which in the setting of G\"odel logics boils down to quantification over all truth values of the underlying truth-value set. Thus, they are somewhere between first order and second order quantifiers. It has already been shown that there are uncountably many quantified propositional G\"odel logics (without first order quantifiers), which is in stark contrast to the fact that there are only countably many first order G\"odel logics. For three cases, namely $V_\infty=[0,1]$, $V_\downarrow = \{0\}\cup\{1/n\mathrel{:} n>0\}$, and $V_\uparrow = \{1\}\cup\{1-1/n\mathrel{:} n>0\}$, quantifier elimination has been shown for the propositional quantified G\"odel logics, in later two cases by extension of the language with an additional operator. In this talk we report on our research program to study the combination of propositional and first-order quantifieres with respect to G\"odel logics.

2010 Mathematics Subject Classification: 03B50, 03C80
Key Words and Phrases: G\"odel Logics, quantified propositional logics, first-order logics

⋅ 5th-E-14:00 − 14:30 Logic for exact real arithmetic (Helmut Schwichtenberg)

Helmut Schwichtenberg, Ludwig Maximilian University of Munich

Previous work with U. Berger, K. Miyamoto and H. Tsuiki is extended in two directions. (1) Instead of viewing the real numbers as abstractly given objects with all the necessary properties assumed as axioms we now use concrete real numbers (Cauchy sequences with moduli) and provide formal proofs in the style of constructive analysis (Bishop 67). Apart from being more complete, this resolves the delicate issue which equality (for reals) is to be used in the clauses of coinductively defined predicates. However, the choice of our model for the reals does not influence the extracted (signed digit or Gray code based) algorithms, since quantifiers over real numbers are taken as non-computational. (2) Using ideas from Ciaffaglione and Gianantonio, we extract signed digit and Gray code based algorithms for division from a proof that the reals are closed under division.

2010 Mathematics Subject Classification: 03D78, 03F60, 03B70, 03B35
Key Words and Phrases: signed digit code, Gray code, real number computation, inductive and coinductive definitions, corecursion, program extraction, realizability

⋅ 5th-E-14:30 − 15:00 Transforming the Weihrauch lattice into a Brouwer algebra (Vasco Brattka, Guido Gherardi)

Vasco Brattka*, Bundeswehr University Munich \& University of Cape Town, Guido Gherardi, University of Bologna

We prove that the Weihrauch lattice can be transformed into a Brouwer algebra by the consecutive application of two closure operators in the appropriate order: first completion and then parallelization. The closure operator of completion is a new closure operator that we introduce. It transforms any problem into a total problem on the completion of the respective types, where we allow any value outside of the original domain of the problem. This closure operator is of interest by itself, as it generates a total version of Weihrauch reducibility that is defined like the usual version of Weihrauch reducibility, but in terms of total realizers. From a logical perspective completion can be seen as a way to make problems independent of their premises. Alongside with the completion operator and total Weihrauch reducibility we need to study precomplete representations hat are required to describe these concepts. In order to show that the parallelized completed Weihrauch lattice forms a Brouwer algebra, we introduce a new multiplicative version of an implication. While the parallelized completed Weihrauch lattice forms a Brouwer algebra with this implication, the completed Weihrauch lattice fails to be a model of intuitionistic linear logic in two different ways. In order to pinpoint the algebraic reasons for this failure, we introduce the concept of a Weihrauch algebra that allows us to formulate the failure in precise and neat terms.

2010 Mathematics Subject Classification: 03B30, 03D30, 03D78, 03F52, 03F60, 06D20
Key Words and Phrases: Weihrauch complexity, computable analysis, Brouwer algebra, intuitionistic and linear logic

⋅ 5th-E-15:00 − 15:30 Formal methods in the software industry - A success story? (Johannes Kanig)

Johannes Kanig, AdaCore

Formal Methods, the use of mathematical techniques for the specification, development and verification of software, have been explored by researchers since the 60ies, as a way to improve software quality. But their application in the software industry has been relatively rare. The reasons for this include perceived complexity of the tools and high cost of their use. In the last decade however, tools have become better, and the software challenges in the industry larger, so that the cost-benefit argument has changed. Large companies are applying formal methods every day to find bugs in their software, or even to prove the absence of bugs. In this talk, we will give an overview over the history of formal methods in the software industry, the changes in the industry that motivate their increased use, and showcase the recent progress and applications of formal methods in the development of software.

2010 Mathematics Subject Classification: 68N30
Key Words and Phrases: formal methods, software development, industry
Advances on automorphic forms and related topics

⋅ 4th-A-09:00 − 10:00   Chair: Youn-Seo Choi (KIAS)

⋅ 4th-A-09:00 − 09:30 Selberg zeta functions with twists of non-expanding cusp monodromies (Anke Pohl)

Anke Pohl, University of Bremen

The Selberg zeta function $Z$ of a hyperbolic surface $X$ is a generating function for the geodesic length spectrum of $X$. It is a deep and influential result that the zeros of $Z$ encode the $L^2$-eigenvalues and resonances of the Laplacian of $X$, thereby showing e.g. an intimate relation between the geometric and spectral properties of the surface $X$. We report on the current status of a project to generalize these results to Selberg zeta function that are twisted by certain non-unitary representations. This is joint work with Ksenia Fedosova.

2010 Mathematics Subject Classification: 11M36, 37C30, 37D35
Key Words and Phrases: Selberg zeta function, non-unitary representation, non-expanding cusp monodromy, transfer operator

⋅ 4th-A-09:30 − 10:00 Waring's problem for rational functions (Bo-Hae Im, Michael Larsen)

임보해*((한국과학기술원)), Michael Larsen((Indiana University))
Bo-Hae Im*, KAIST, Michael Larsen, Indiana University

We introduce various generalized versions of the classical Waring's problem which asks whether there exists a positive integer $N(d)$ such that every positive integer can be written as the sum of $N(d)$ $d$th powers of positive integers. In particular we talk about Waring's problem for rational functions in one variable. This is a joint work with Michael Larsen.

2010 Mathematics Subject Classification: 11P05
Key Words and Phrases: Waring's problem

⋅ 4th-B-13:30 − 14:30   Chair: Byeong-Kweon Oh (Seoul National University)

⋅ 4th-B-13:30 − 14:00 Computation of Belyi maps and Hurwitz families of Galois covers (Joachim Koenig, Dominik Barth, Andreas Wenz)

Joachim Koenig*((한국과학기술원)), Dominik Barth((University of Wuerzburg)), Andreas Wenz((University of Wuerzburg))
Joachim Koenig*, KAIST, Dominik Barth, University of Wuerzburg, Andreas Wenz, University of Wuerzburg

We review some recently developed techniques for the computation of high-degree Belyi maps via modular forms on Fuchsian groups. We combine these techniques with deformation methods to obtain algorithms for the explicit computation of Hurwitz spaces and parametric families of Galois covers. Applications include the first explicit Galois realizations of several almost-simple groups of large permutation degree, as well as the first totally real Galois realizations for several further groups.

2010 Mathematics Subject Classification: 11R32
Key Words and Phrases: Galois theory, coverings, Belyi maps, modular forms, explicit computation

⋅ 4th-B-14:00 − 14:30 An example of real quadratic field with finite nonsolvable maximal unramified extension (Kwang-Seob Kim)

김광섭((조선대))
Kwang-Seob Kim, Chosun University

Under the assumption of the generalized Riemann hypothesis (GRH), we show that there is a real quadratic field K such that the etale fundamental group of the spectrum of the ring of integers of K is isomorphic to A5. The proof uses standard methods involving Odlyzko bounds, as well as the proof of Serre's modularity conjecture. To the best of the author's knowledge, this is the first example of a number field K for which etale fundamental group of the spectrum of the ring of integers of K is finite, nonabelian and simple under the assumption of the GRH.

2010 Mathematics Subject Classification: 11R29
Key Words and Phrases: nonabelian etale fundamental groups, nonabelian simple unramified extensions of number fields, Serre's modularity conjecture, Class number one problems

⋅ 4th-B-14:30 − 15:30   Chair: Chang Heon Kim (Sungkyunkwan University)

⋅ 4th-B-14:30 − 15:00 Classification of discrete series representations of odd $GSpin$ groups and its generalization (Yeansu Kim)

김연수((전남대))
Yeansu Kim, Chonnam National University

The classification of discrete series is one important subject with numerous application in the harmonic analysis and in the theory of automorphic forms. We explain the result on the classification of discrete series of odd general spin groups, generalizing the M\oe glin-Tadi\'c classification for classical groups. If time permits, I will explain how our approach can be generalized to special cases of other connected reductive groups. This is the joint project with Ivan Mati\'c.

2010 Mathematics Subject Classification: 22E35, 22E50, 11F70
Key Words and Phrases: discrete series

⋅ 4th-B-15:00 − 15:30 On the Ramanujan-Petersson conjecture for modular forms of half-integral weight (Winfried Kohnen)

Winfried Kohnen, Heidelberg University

We will discuss several aspects of a ``Ramanujan-Petersson conjecture" for the growth of the Fourier coefficients of cusp forms of half-integral weight.

2010 Mathematics Subject Classification: 11F37
Key Words and Phrases: modular form, half-integral weight

⋅ 5th-E-13:30 − 14:30   Chair: Jaebum Sohn (Yonsei University)

⋅ 5th-E-13:30 − 14:00 Higher Green functions and their CM values (Jan Hendrik Bruinier, Stephan Ehlen, Tonghai Yang)

Jan Hendrik Bruinier, Technical University of Darmstadt, Stephan Ehlen*, University of Cologne, Tonghai Yang, University of Wisconsin-Madison

In my talk I will present results from joint work in progress with Jan Bruinier and Tonghai Yang. We construct higher Green functions on orthogonal Shimura varieties of signature $(n,2)$ using a regularized theta lift. We also consider their special values at certain CM points and prove algebraicity results for these special values.

2010 Mathematics Subject Classification: 11F37, 11G15
Key Words and Phrases: higher Green's functions, CM values

⋅ 5th-E-14:00 − 14:30 Computations of the orders of certain cuspidal divisors on modular Jacobian varieties (Hwajong Yoo)

유화종((서울대))
Hwajong Yoo, Seoul National University

Let $J_0(N)$ be the Jacobian variety of the modular curve $X_0(N)$. Let $C(N)$ be the cuspidal group of $J_0(N)$. As Ohta studied, the odd part of $C(N)$ is decomposed into eigenspaces of Atkin-Lehner involutions when $N$ is squarefree. In this talk, we explicitly describe generators of these eigenspaces and compute the precise orders of them. More generally, for a squarefree integer $N$ we compute the precise order of the image in $J_0(N)$ of any degree-0 cuspidal divisor on $X_0(N)$.

2010 Mathematics Subject Classification: 11G16, 11G18
Key Words and Phrases: cuspidal groups

⋅ 5th-E-14:30 − 15:30   Chair: Soon-Yi Kang (Kangwon National University)

⋅ 5th-E-14:30 − 15:00 On the Iwasawa main conjecture for elliptic curves (Chan-Ho Kim)

김찬호((고등과학원))
Chan-Ho Kim, KIAS

For a given rational elliptic curve $E$ and an odd prime $p$, we explain how the Iwasawa main conjecture for the pair $(E,p)$ can be numerically verified under mild assumptions. In particular, this method is insensitive to the reduction type of elliptic curves. This talk is based on the joint work with Myoungil Kim and Hae-Sang Sun and that with Kentaro Nakamura.

2010 Mathematics Subject Classification: 11R23, 11F67
Key Words and Phrases: elliptic curves, modular forms, Iwasawa theory

⋅ 5th-E-15:00 − 15:30 On the Kudla-Rapoport conjecture (Sungmun Cho)

조성문((포항공대))
Sungmun Cho, POSTECH

Kudla-Rapoport conjecture predicts a relation between the local intersection multiplicity for special cycles on the Rapoport-Zink space for $GU(1,n-1)$ and (the derivative of) the hermitian Siegel series. In this talk, I will suggest a theoretical conjecture related to this for any n.

2010 Mathematics Subject Classification: 11F30, 11F46, 11G18, 14C17, 14G35, 14J15
Key Words and Phrases: Kudla-Rapoport conjecture, hermitian Siegel series, hermitian Gross-Keating invariant
Symplectic Geometry

⋅ 4th-A-09:00 − 10:00   Chair: Urs Frauenfelder (Seoul National University)

⋅ 4th-A-09:00 − 09:30 Equivariant wrapped Floer homology and symmetric periodic orbits (Joontae Kim, Seongchan Kim, Myeonggi Kwon)

김준태*((고등과학원)), 김성찬((University of Augsburg)), 권명기((University of Giessen))
Joontae Kim*, KIAS, Seongchan Kim, University of Augsburg, Myeonggi Kwon, University of Giessen

We introduce equivariant wrapped Floer homology using an anti-symplectic involution, which is useful to study symmetric periodic Reeb orbits. With careful analysis on index iterations, we give the minimal number of geometrically distinct symmetric periodic Reeb orbits on real contact manifolds. The most interesting examples are star-shaped hypersurfaces in the complex plane with a natural involution, namely, the complex conjugate. This is joint work with Seongchan Kim and Myeonggi Kwon.

2010 Mathematics Subject Classification: 53D40, 37C27, 53D12
Key Words and Phrases: Floer homology, symmetric periodic Reeb orbit, index iteration

⋅ 4th-A-09:30 − 10:00 On the existence and non-existence of global surfaces of section (Otto van Koert)

Otto van Koert((서울대))
Otto van Koert, Seoul National University

A global surface of section in a 3-manifold with a flow is an embedded surface with the property that, except for the boundary of the surface, every orbit of the flow intersects the surface both in future and past time. This notion was developed by Poincar\'e and Birkhoff and can be used to simplify the dynamics: instead of flows one can study diffeomorphisms of surfaces. We will survey some existence results of global surfaces of section before proceeding to several non-existence results, and give some dynamical consequences.

2010 Mathematics Subject Classification: 37J99, 53D35
Key Words and Phrases: global surfaces of section

⋅ 4th-B-13:30 − 14:30   Chair: Cheol-Hyun Cho (Seoul National University)

⋅ 4th-B-13:30 − 14:00 A wrapped Fukaya category of knot complement and hyperbolic knots (Youngjin Bae, Seonhwa Kim, Yong-Geun Oh)

배영진((Kyoto University)), 김선화((기초과학연구원)), 오용근*((기초과학연구원))
Youngjin Bae, Kyoto University, Seonhwa Kim, IBS-CGP, Yong-Geun Oh*, IBS-CGP

In this talk, we will present a new construction of wrapped Fukaya category of knot-complement $S^3 \setminus K$. This is a version of wrapped Fukaya category of the cotangent bundle $T^*(S^3 \setminus K)$ of the knot-complement $S^3 \setminus K$, which is noncompact. We will explain how to overcome noncompactness of the base which requires a careful application of the strong maximum principle to handle the C0 bound in the horizontal direction. We also present a formality result for the $A_\infty$ structure of the Floer complex of the conormal of the `ideal boundary' of for hyperbolic knot complement, and prove $m_k = 0$ for all except $k = 2$ for the case.

2010 Mathematics Subject Classification: 53D37, 57M25
Key Words and Phrases: knot complement, wrapped Fukaya category, hyperbolic knots

⋅ 4th-B-14:00 − 14:30 What might a Hamiltonian delay equation be (Peter Albers, Urs Frauenfelder, Felix Schlenk)

Peter Albers, Heidelberg University, Urs Frauenfelder*, University of Augsburg, Felix Schlenk, University of Neuchatel

This is joint work with Peter Albers and Felix Schlenk. Although we do not know what a Hamiltonian delay in general is, we describe a variational approach to their periodic solutions. I will explain how periodic solutions of some delayed Lotka-Volterra equations arise in this way. Finally I will discuss how this leads to some new directions and challenges for Floer homology.

2010 Mathematics Subject Classification: 34K, 58E05, 58F05, 70K42
Key Words and Phrases: delay equation, Hamiltonian system, action functional

⋅ 4th-B-14:30 − 15:30   Chair: Yong-Geun Oh (POSTECH)

⋅ 4th-B-14:30 − 15:00 Counting holomorphic curves with jet conditions (Klaus Mohnke)

Klaus Mohnke, Humboldt University of Berlin

What is the number of elements counted with sign of the set of holomorphic maps $u:\mathbb{C}P^1\to \mathbb{C}P^n$ modulo reparametrizations of degree $d$ with $(\varphi\circ u)^{(k)} = 0$ for a given holomorphic function on a neighbourhood of $0\in \mathbb{C}P^n$ and $k=0,\ldots,d(n+1)-2$? For $d=1$ and $d=2$ the answer will be explained already indicating that it might be not so easy to obtain.

2010 Mathematics Subject Classification: 53D45
Key Words and Phrases: holomorphic spheres, jet condition

⋅ 4th-B-15:00 − 15:30 Real Lagrangians in Brieskorn Milnor fibers (Myeonggi Kwon)

Myeonggi Kwon, University of Giessen

Brieskorn Milnor fibers are open symplectic manifolds defined by regular level sets of Brieskorn-type complex polynomials. In this talk, we define a family of anti-symplectic involutions on them, which are essentially given by the complex conjugation, and we consider real Lagrangians taking their fixed point set. We discuss topological properties and wrapped Floer homology of such real Lagrangians. This talk is based on joint work with Joontae Kim.

2010 Mathematics Subject Classification: 53D40, 53D12
Key Words and Phrases: Brieskorn Milnor, fibers real Lagrangians, wrapped Floer homology

⋅ 5th-E-13:30 − 14:30   Chair: Klaus Mohnke (Humboldt University of Berlin)

⋅ 5th-E-13:30 − 14:00 Exotic symplectic dynamics on $S^3$ (Kai Zehmisch)

Kai Zehmisch, University of Giessen

Exotic symplectic dynamics on $S^3$ A Hamiltonian version of Gottschalk's conjecture asks for non-minimality of characteristic flows of an odd-symplectic form on $S^3$. In my talk I will indicate a construction of exotic symplectic flows whose defining odd-symplectic form extends to a symplectic form on the 4-ball; but minimality properties for these flows as well as properties of Fish-Hofer's feral holomorphic curves are not known.

2010 Mathematics Subject Classification: 37J05, 53D35
Key Words and Phrases: odd-symplectic form, symplectic cobordism, minimal flow

⋅ 5th-E-14:00 − 14:30 A Chekanov-Eliashberg algebra for Legendrian graphs (Youngjin Bae, Byung Hee An)

배영진*((Kyoto University)), 안병희((기초과학연구원))
Youngjin Bae*, Kyoto University, Byung Hee An, IBS-CGP

We define a differential graded algebra for Legendrian graphs in the standard contact Euclidean three space. This invariant is defined combinatorially by using ideas from the bordered version of Legendrian contact homology. A set of countably many generators and a generalized notion of equivalence are introduced for invariance. I will talk about a van Kampen type theorem for the DGA and a relation to a partially wrapped Floer homology of an induced Liouville sector. This is a joint work with Byung Hee An.

2010 Mathematics Subject Classification: 53D42, 57R17, 57M15
Key Words and Phrases: Chekanov-Eliashberg algebra, Legendrian contact homology, Legendrian graphs, Bordered Legendrians

⋅ 5th-E-14:30 − 15:30   Chair: Jungsoo Kang (Seoul National University)

⋅ 5th-E-14:30 − 15:00 Monotone Lagrangians in flag varieties (Yunhyung Cho, Yoosik Kim, Yong-Geun Oh)

조윤형*((성균관대)), 김유식((Boston University)), 오용근((포항공대 \& 기초과학연구원))
Yunhyung Cho*, Sungkyunkwan University, Yoosik Kim, Boston University, Yong-Geun Oh, POSTECH \& IBS-CGP

In this talk, I will explain certain monotone Lagrangians appeared as fibers of certain completely integrable systems and discuss the (non-) displaceability using disc potentials. As a by-product, we can produce a family of non-displaceable monotone Lagrangian tori in cotangent bundles of certain manifolds, including n-spheres. This is joint work with Yoosik Kim and Yong-Geun Oh.

2010 Mathematics Subject Classification: 53D05
Key Words and Phrases: symplectic manifold, Lagrangian submanifold, potential function

⋅ 5th-E-15:00 − 15:30 Two results on Zoll structures on $S^2$ (Stefan Suhr)

Stefan Suhr, Ruhr-University Bochum

I will present two results, one in cooperation with Urs Frauenfelder and Christian Lange and one in cooperation with Marco Mazzucchelli. Gromoll \& Grove proved that every Besse Riemannian metric on $S^2$ is already Zoll. I will explain a generalization to real Hamiltonian structures on $\mathbb R P^3$. The main step is a characterization of orbifold quotients of $\mathbb R P^3$. The second result characterizes Zoll metrics on $S^2$ by the coincidence of the Lusternik-Schnirelman critical values on the space of unoriented embeddings of $S^1$ into $S^2$. The main step is the construction of a nonvanishing cohomology class on a neighborhood of the simple closed geodesics.

2010 Mathematics Subject Classification: 53C22, 53D35, 53D25, 58E10
Key Words and Phrases: real hamiltonian structures, zoll metrics
Quantum attack models and post quantum cryptography

⋅ 4th-A-09:00 − 10:00   Chair: Sang Geun Hahn (KAIST)

⋅ 4th-A-09:00 − 09:30 CCA-secure key encapsulation mechanism based on binary LWE (Minhye Seo, Suhri Kim, Dong Hoon Lee, Jong Hwan Park)

서민혜*((고려대)), 김수리((고려대)), 이동훈((고려대)), 박종환((상명대))
Minhye Seo*, Korea University, Suhri Kim, Korea University, Dong Hoon Lee, Korea University, Jong Hwan Park, Sangmyung University

By remarkable progress in quantum computing, interest in quantum-resistant cryptography has increased, and the National Institute of Standards and Technology (NIST) has begun a process to develop post-quantum cryptography standardization. Lattice-based cryptography is one of the promising candidates of post-quantum cryptography, which provides higher security based on the worst-case hardness of lattice problems. In this paper, we propose new key encapsulation mechanisms (KEMs), EMBLEM and R.EMBLEM, based on the learning with errors (LWE) problem. First, we propose a new alternative of message encoding for LWE-based encryption schemes. Our encoding method is a new approach in that it does not require any rounding operation (for comparison) in the decryption phase. And this allows us to more intuitively extend to multi-bit encoding. We also adopt the binary LWE to reduce the overall computational complexity. In other words, by using the binary (\textit{or} small secret) LWE, we can encode multiple bits at a time without increasing the size of modulus $q$. We set parameters so that our proposed KEMs have a negligible probability of decryption failure, approximately $2^{-140}$, which allows them to meet the security requirement by NIST. We then provide the implementation results of EMBLEM and R.EMBLEM, and evaluate their performances compared to other LWE-based KEMs in the literature.

2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: lattice-based cryptography, key encapsulation mechanism, binary LWE

⋅ 4th-A-09:30 − 10:00 Sieving algorithms for the shortest vector problem (Elena Kirshanova)

Elena Kirshanova, ENS Lyon

In this talk we give an overview on the complexity of the algorithms for the Shortest Vector Problem (SVP) with the focus on a particular family of algorithms: sieving algorithms. First, we explain why this problem is important and how the complexity of SVP algorithms impacts parameters for cryptographic protocols. Then, we present recent advances in memory efficient versions of sieving algorithms. Time permitting, we touch upon locality-sensitive techniques for these type of algorithms. The talk is based on joint works with Gottfried Herold and Thijs Laarhoven.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: lattice, shortest vector problem, sieving algorithms

⋅ 4th-B-13:30 − 15:30   Chair: Jooyoung Lee (KAIST)

⋅ 4th-B-13:30 − 14:00 Symmetric encryption adapted to fully homomorphic encryption: Investigating new criteria on Boolean functions (Pierrick M\'eaux)

Pierrick M\'eaux, Catholic University of Louvain

Fully Homomorphic Encryption is a recent powerful cryptographic construction, which enables one to securely compute all functions on encrypted data, and decrypt the result of the function applied to the real data. This construction gives the possibility to securely delegate computation, which is a very important property with the increasing development of Cloud computing. Nevertheless, in current client-server frameworks, the client devices are too restricted to support pure FHE. In order to solve this problem, FHE has to be combined with primitives which incur small computation and communication cost: Symmetric Encryption schemes. Traditional SE schemes are not suited to be combined with FHE, therefore new SE schemes with low decryption complexity are considered, as the family of stream ciphers FLIP. In this construction, the low complexity makes the security crucially depend on the properties of a Boolean function, which should be robust relatively to standard cryptographic criteria and new ones. In this talk, we will present some new criteria on Boolean function it makes arise, and families of Boolean functions with good parameters relatively to these criteria.

2010 Mathematics Subject Classification: 06E10, 68P25, 68P30, 94A60
Key Words and Phrases: cryptography, Boolean functions, fully homomorphic encryption

⋅ 4th-B-14:00 − 14:30 Quantum search and related problems (Aaram Yun)

윤아람((울산과학기술원))
Aaram Yun, UNIST

It is well-known that Grover's algorithm is the optimal among quantum search algorithms in all regimes. We may also consider the problem of deciding whether an oracle is the constant zero function or a random delta function, and for this, previous known bound $O(q/\sqrt{N})$ was larger than that achievable by Grover's algorithm, which is $O(q^2/N)$. I will show that in fact the tighter bound $O(q^2/N)$ holds. Also, I will present problems closely related to this, namely, one-way-to-hiding lemma and oracle reprogramming, and discuss whether similar tight bound can be obtained for these.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Grover's algorithm, quantum search, quantum random oracle model, oracle reprogramming, one-way-to-hiding

⋅ 4th-B-14:30 − 15:00 New applications of the lossy mode of LWE (Benoit Libert)

Benoit Libert, CNRS \& ENS Lyon

Given a random narrow matrix $\mathbf{A} \in \mathbb{Z}_q^{m \times n}$, the Learning-With-Errors (LWE) problem is to distinguish uniformly random vectors over $\mathbb{Z}_q^m$ from vectors of the form $\mathbf{A} \cdot \mathbf{s} + \mathbf{e}$, for some secret $\mathbf{s} \in \mathbb{Z}^n$ and some small-norm noise vector $\mathbb{e} \in \mathbb{Z}^m$. While the LWE function $(\mathbf{s},\mathbf{e}) \rightarrow \mathbf{A} \cdot \mathbf{s} + \mathbf{e}$ can be injective when the matrix $\mathbf{A}$ is random, it may induce a lossy function when the columns of $\mathbf{A}$ are close to the column-span of a matrix $\mathbf{B} \in \mathbb{Z}_q^{m \times \ell}$ of lower rank $\ell < n$. Yet, the injective and lossy modes are computationally indistinguishable under the LWE assumption in dimension $\ell$. This talk will present new recent applications of the lossy mode of the LWE function. The first one is the design of a lattice-based public-key encryption scheme with \linebreak simulation-based selective-opening security (which preserves the security of un-opened ciphertexts when some senders are corrupted) under chosen-ciphertext attacks. The second application is the construction of non-interactive distributed pseudo-random functions in the adaptive corruption setting.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: LWE, lossy trapdoor functions, selective opening security, pseudo-random functions

⋅ 4th-B-15:00 − 15:30 On McNie, a new code-based public-key cryptosystem (Jon-Lark Kim, Young-Sik Kim, Lucky Galvez, Myeong Jae Kim, Nari Lee)

김종락*((서강대)), 김영식((조선대)), Lucky Galvez((서강대)), 김명재((서강대)), 이나리((서강대))
Jon-Lark Kim*, Sogang University, Young-Sik Kim, Chosun University, Lucky Galvez, Sogang University, Myeong Jae Kim, Sogang University, Nari Lee, Sogang University

In this talk, we describe a new code-based public key encryption scheme, called McNie. McNie is a hybrid version of the McEliece and Niederreiter cryptosystems. McNie passed the 1st round of the NIST competition on Post-Quantum Cryptography. The security of McNie is reduced to the hardness of syndrome decoding. The public key involves a random generator matrix which is also used to mask the private code. This makes the system safer against known structural attacks. In particular, we apply rank-metric codes to McNie. We also discuss recent attacks on McNie.

2010 Mathematics Subject Classification: 14G50, 81P94
Key Words and Phrases: McEliece cryptosystem, McNie, Niederreiter cryptosystem, public-key encryption

⋅ 5th-E-13:30 − 15:00   Chair: Kyung-Ah Shim (NIMS)

⋅ 5th-E-13:30 − 14:00 HiMQ-3: A high speed signature scheme based on multivariate quadratic equations (Kyung-Ah Shim, Cheol Min Park, Namhun Koo, Aeyoung Kim)

심경아((국가수리과학연구소)), 박철민*((국가수리과학연구소)), 구남훈((성균관대)), 김애영((한양대))
Kyung-Ah Shim, NIMS, Cheol Min Park*, NIMS, Namhun Koo, Sungkyunkwan University, Aeyoung Kim, Hanyang University

A multivariate quadratic Public-Key Cryptography (MQPKC) is one of the most pro\-mising alternatives for classical PKCs after the eventual coming of quantum computers. In this work, we propose a new signature scheme, HiMQ-3, based on multivariate quadratic equations with short secret key and fast performance. Our signature scheme use solvable quadratic equations with chain cross terms and consecutive cross terms unlike other MQ-signatures based on the Oil-Vinegar method. For high speed implementation and the shortest secret key size, we also design sparse version of our scheme, HiMQ-3S. We provide a direct comparison of implementation results for our scheme, Rainbow and classical ones such as RSA, ECDSA on the same platform for secure and optimal parameters at a 128-bit security level. A proposal for HiMQ-3 was submitted to the US National Institute of Standards and Technology (NIST) Post-Quantum Cryptography Project.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: post-quantum cryptography, multivariate-quadratic scheme, signature scheme

⋅ 5th-E-14:00 − 14:30 Sieving in practice (Martin Albrecht, Leo Ducas, Gottfried Herold, Elena Kirshanova, Eamonn Postlethwaite, Marc Stevens)

Martin Albrecht, Royal Holloway University of London, Leo Ducas, CWI Amsterdam, Gottfried Herold*, ENS Lyon, Elena Kirshanova, ENS Lyon, Eamonn Postlethwaite, Royal Holloway University of London, Marc Stevens, CWI Amsterdam

It is a common belief that algorithms requiring large amount of space are not practical. Sieving methods for the Shortest Vector Problem (SVP), being memory inefficient, are good examples of such algorithms: the largest SVP challenges that have been solved were solving using other memory-friendly algorithms. In this talk we present an on-going joint work with Martin Albrecht, Leo Ducas, Elena Kirshanova, Eamonn Postlethwaite and Marc Stevens on implementation of sieving algorithms for the Shortest Vector Problem. We describe technical challenges that arise when one tries to make sieving algorithms practical, and how we can overcome some of them.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: lattice algorithms, shortest vector problem, sieving

⋅ 5th-E-14:30 − 15:00 Improved bootstrapping techniques for lattice-based fully homomorphic encryption (Hao Chen, Kyoohyung Han)

Hao Chen((Microsoft Research)), 한규형*((서울대))
Hao Chen, Microsoft Research, Kyoohyung Han*, Seoul National University

Bootstrapping is a crucial operation in Gentry's breakthrough work on fully homomorphic encryption (FHE), where a homomorphic encryption scheme evaluates its own decryption algorithm. There has been a couple of implementations of bootstrapping, among which HElib arguably marks the state-of-the-art in terms of throughput, ciphertext/message size ratio and support for large plaintext moduli. In this work, we applied a family of ``lowest digit removal" polynomials to design an improved homomorphic digit extraction algorithm which is a crucial part in bootstrapping for both FV and BGV schemes. When the secret key has 1-norm $h = ||s||_1$ and the plaintext modulus is $t = p^r$, we achieved bootstrapping depth $\log{h}+\log(\log_{p}(ht))$ in FV scheme. In case of the BGV scheme, we brought down the depth from $\log{h}+2 \log{t}$ to $\log{h}+\log{t}$. We implemented bootstrapping for FV in the SEAL library. We also introduced another slim mode which restricts the plaintexts to batched vectors in $\mathbb{Z}_{p^r}$. The slim mode has similar throughput as the full mode, while each individual run is much faster and uses much smaller memory. For example, bootstrapping takes 6.75 seconds for vectors over $\text{GF}(127)$ with 64 slots and 1381 seconds for vectors over $\text{GF}(257^128)$ with 128 slots.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: cryptography, homomorphic encryption, bootstrapping
Advances in Computational PDEs

⋅ 4th-A-09:00 − 10:00   Chair: Eun-Jae Park (Yonsei University)

⋅ 4th-A-09:00 − 09:30 Nonconforming finite element methods for multiscale problems (Dongwoo Sheen)

신동우((서울대))
Dongwoo Sheen, Seoul National University

In this talk we apply nonconofrming finite elements to solve multiscale problems. First, we present a general framework for nonconforming multiscale approach based on the GMsFEM (Generalized Multiscale Finite Element Methods). Five Steps for the nonconforming GMsFEM will be given explicitly. Next, we present nonconforming finite element method for the heterogeneous multiscale method (HMM). For both methods, error estimates are derived and differences between conforming and nonconforming approaches for solving multiscale methods. Numerical results will be shown and discussed.

2010 Mathematics Subject Classification: 65N30
Key Words and Phrases: nonconforming finite element, multiscale, homogenization

⋅ 4th-A-09:30 − 10:00 Weakly symmetric stress reconstruction and a posteriori error estimation for linear elasticity (Fleurianne Bertrand)

Fleurianne Bertrand, University of Duisburg-Essen

In this talk, a stress reconstruction in the H(div)-conforming Raviart-Thomas space based on the Taylor-Hood displacement-pressure approximation procedure for linear elasticity is proposed. The construction is weakly symmetric and the computation is performed locally on a set of vertex patches. Due to the weak symmetry constraint, the local problems need to satisfy consistency conditions associated with all rigid body modes. The resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient.

2010 Mathematics Subject Classification: 65N30, 65N50
Key Words and Phrases: a posteriori error estimation, incompressible linear elasticity, Taylor-Hood elements, weakly symmetric stress equilibration, Raviart-Thomas elements

⋅ 4th-B-13:30 − 15:30   Chair: Dong-wook Shin (NIMS)

⋅ 4th-B-13:30 − 14:00 Domain decomposition preconditioners for elliptic problems with highly varying and random coefficients (Hyea Hyun Kim, Eric Chung, Junxian Wang)

김혜현*((경희대)), Eric Chung ((The Chinese University of Hong Kong)), Junxian Wang ((Xiangtan University))
Hyea Hyun Kim*, Kyung Hee University, Eric Chung, The Chinese University of Hong Kong, Junxian Wang, Xiangtan University

Domain decomposition preconditioners are useful for enhancing the convergence of an iterative method for numerical partial differential equations. Efficient preconditioners are often consist of two ingredients, i.e., one coarse problem, and many subdomain problems. Introduction of more subdomain problems can reduce the local problem size so that the cost for solving local problems can become cheaper. On the other hand, the subdomain problems can only correct residual errors locally and it makes the convergence of iterative methods slower. Another ingredient, the coarse problem is thus introduced to correct the global residual errors. The design of the coarse problem is important in obtaining the robustness of the preconditioners to the number of subdomains, i.e., effectively correct the global residual errors. For a model problem with highly varying and random coefficients, elaborated construction of coarse basis functions is necessary to find an effective coarse problem. Certain eigenvalue problems are formed on each interface of subdomains or on each subdomains to find suitable coarse basis functions. Estimate of conditioner numbers for the proposed preconditioner is shown. Numerical results are presented to confirm the performance of the proposed method for highly varying and random coefficient model problems.

2010 Mathematics Subject Classification: 65F08
Key Words and Phrases: iterative method, preconditioner, domain decomposition methods, condition numbers, eigenvalue problems, coarse problem

⋅ 4th-B-14:00 − 14:30 Adaptive least-squares finite element methods (Carsten Carstensen)

Carsten Carstensen, Humboldt University of Berlin

The least-squares functional is a reliable and ecient error estimator with global upper and lower bounds and can be very accurate. The elementwise contributions to the global L2 norm serves well as renement indicators in adaptive mesh-rening algorithms but the convergence analysis is less well understood. Those local contributions do not involve an explicit mesh-size factor and hence their reduction is unclear. The paper Bringmann, Carstensen, and Park (Numer. Math. 2017) guarantees the plain convergence for a bulk parameter close to one and that is far away from the arguments for rate optimality. The axioms of adaptivity in Carstensen, Feischl, Page, and Praetorius (Comp. Math. Appl. 2014) are not available and an alternative error estimator is derived and exploited in Carstensen and Park (SIAM J. Numer. Anal. 2015) and enforces a separate marking strategy with an overall abstract theory by Carstensen and Rabus (SIAM J. Numer. Anal. 2017). The presentation discusses on all those aspects for the Laplace, the Stokes and the Lame-Navier equations as in Bringmann and Carstensen (Numer. Math. 2017). Numerical experiments confirm the proven optimal convergence rates.

2010 Mathematics Subject Classification: 65N30
Key Words and Phrases: adaptivity, optimal rates, least-squares

⋅ 4th-B-14:30 − 15:00 The hybrid difference methods for the wave equations on exterior domains (Youngmok Jeon)

전영목((아주대))
Youngmok Jeon, Ajou University

In this presentation we introduce the hybrid differences(HDM) for the static and time- dependent wave equations and a new absorbing boundary condition, so called the discrete radial absorbing boundary condition (ABC). The HDM is a finite difference version of the hybridized discontinuous Galerkin method, and it is simple to be implemented and has many advantages compared to the existing finite difference methods. The discrete radial ABC is obtained by directly invoking to the multipole expansion of the wave solutions, and it is in an algebraic form, which annihilates the first several terms of multipole expansions exactly. The idea stems from that of Bayless-Turkel, and they induce ABCs in a differential operator form. The new ABC is degenerated to a quasi-Dirichlet condition, where the Dirichlet condition is replaced by the time-extrapolation of data in the previous time steps. Numerical examples confirming the efficiency of the new ABC and HDM are presented.

2010 Mathematics Subject Classification: 65N30, 65N38, 65N50
Key Words and Phrases: Crank Nicholson, discrete radial absorbing boundary condition, hybrid difference, wave equation

⋅ 4th-B-15:00 − 15:30 Maximum-norm a posteriori error bounds for parabolic problems (Torsten Lin{\ss})

Torsten Lin{\ss}, University of Hagen

For classical and singularly perturbed parabolic equations, we present maximum norm a posteriori error estimates that, in the singularly perturbed regime, hold uniformly in the small perturbation parameter. The parabolic equations are discretised in time using the backward Euler method, the Crank-Nicolson method and the discontinuous Galerkin dG(r) method. Both semidiscrete (no spatial discretation) and fully discrete cases will be considered. The analysis invokes elliptic reconstructions and elliptic a posteriori error estimates. Joint work with Natalia Kopteva (University of Limerick, Ireland) \begin{thebibliography}{9} \bibitem{MR3032709} N. Kopteva and T. Lin{\ss}, {\it Maximum norm a posteriori error estimation for a time-dependent reaction-diffusion problem}, Comput. Methods Appl. Math. {\bf 12} (2012), no. 2, 189--205. \bibitem{MR3056758} N. Kopteva and T. Lin{\ss}, {\it Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions}, SIAM J. Numer. Anal. {\bf 51} (2013), no. 3, 1494--1524. \bibitem{MR3149972} N. Kopteva and T. Lin{\ss}, {\it Numerical study of maximum norm a posteriori error estimates for singularly perturbed parabolic problems}, In Numerical analysis and its applications, volume 8236 of Lecture Notes in Comput. Sci., pages 50--61. Springer, Heidelberg, 2013. \bibitem{MR3720388} N. Kopteva and T. Lin{\ss}, {\it Improved maximum-norm a posteriori error estimates for linear and semilinear parabolic equations}, Adv. Comput. Math. {\bf 43} (2017), no. 5, 999--1022. \end{thebibliography}

2010 Mathematics Subject Classification: 65M60
Key Words and Phrases: parabolic problems, a posteriori error bounds

⋅ 5th-E-13:30 − 15:30   Chair: Youngmok Jeon (Ajou University)

⋅ 5th-E-13:30 − 14:00 A high order discontinuous Galerkin method with skeletal multipliers for convection-diffusion-reaction problems (Mi-Young Kim, Dong-wook Shin)

김미영*((인하대)), 신동욱((연세대))
Mi-Young Kim*, Inha University, Dong-wook Shin, Yonsei University

A discontinuous Galerkin method with skeletal multipliers (DGSM) is developed for diffusion problem. Skeletal multiplier is introduced on the edge/face of each element through the definition of a weak divergence and weak derivative in the method. The local weak formulation is derived by weakly imposing the Dirichlet boundary condition and continuity of fluxes and solutions on the edges/faces. The global weak formulation is then obtained by adding all the local problems. Equivalence of the weak formulation and the original problem is proved. Stability of DGSM is shown and an error estimate is derived in a broken norm. A DGSM for linear convection-diffusion-reaction problems is also derived. An explanation on algorithmic aspects is given. Some numerical results are presented. Singularities due to discontinuities in the diffusion coefficients are accurately approximated. Internal/boundary layers are well captured without showing spurious oscillations. Robustness of the method in increasingly small diffusivity is demonstrated on the whole domain.

2010 Mathematics Subject Classification: 65N12
Key Words and Phrases: convection-diffusion-reaction problem, weak divergence, weak derivative, discontinuous Galerkin method, hybridizable discontinuous Galerkin method, weak Galerkin method

⋅ 5th-E-14:00 − 14:30 Contraction property of an adaptive Hdiv-DG method for the Stokes problem (Guido Kanschat)

Guido Kanschat, Heidelberg University

We present pressure free error estimators for cochain-based discretizations of the Stokes problem. For these estimators, we show strict contraction of adaptive methods and quasi-optimality of the resulting meshes. In addition, we discuss efficient multigrid solvers.

2010 Mathematics Subject Classification: 65N30
Key Words and Phrases: Stokes equations, adaptive finite elements, finite element cochain complex, discontinuous Galerkin

⋅ 5th-E-14:30 − 15:00 Enhancing the mixed finite element eigenvalue approximation (Kwang-Yeon Kim)

김광연((강원대))
Kwang-Yeon Kim, Kangwon National University

In this talk we present two postprocessing schemes which construct some higher order $H^1$-conforming eigenfunction approximations for the Raviart--Thomas mixed finite elements. The purpose of these schemes is two-fold. First, they are used to improve the convergence rate of the eigenvalue approximation via the well-known Rayleigh quotient formula. Second, they lead to asymptotically exact error estimators for the $L^2$ errors of the eigenfunctions. In the first scheme we solve a global primal source problem using a higher order conforming space. It is shown that a higher order eigenfunction approximation can be obtained even on non-uniform meshes by virtue of the superconvergence result of the scalar eigenfunction approximation. The second scheme is local in nature as the main computation lies in solving local source problems. But it relies on a superconvergent vector approximation and thus applies only to low-order elements. Besides a higher convergence rate is ensured for mildly structured meshes.

2010 Mathematics Subject Classification: 65N25, 65N30
Key Words and Phrases: Raviart--Thomas mixed finite element, eigenvalue problem, postprocessing, a posteriori error estimator

⋅ 5th-E-15:00 − 15:30 Optimal adaptive algorithms for indefinite problems (Michael Feischl)

Michael Feischl, Karlsruhe Institute of Technology

We develop a framework which allows us to prove the essential general quasi-orthogo\-nality for non-symmetric and indefinite problems as the stationary Stokes problem or certain transmission problems. General quasi-orthogonality is a necessary ingredient of rate optimality proofs and is the major difficulty on the way to prove rate optimal convergence of adaptive algorithms for many strongly non-symmetric or indefinite problems. The proof exploits a new connection between the general quasi-orthogonality and LU-factorization of infinite matrices.

2010 Mathematics Subject Classification: 65M50, 65M60
Key Words and Phrases: adaptive algorithms, error estimator, optimal rates
Algebraic geometry and computer vision

⋅ 5th-D-09:00 − 10:00   Chair: Donghoon Hyeon (Seoul National University)

⋅ 5th-D-09:00 − 09:30 [CANCELLED] Efficiently computing the Betti numbers of symmetric semi-algebraic sets (Cordian Riener)

Cordian Riener, Arctic University of Norway

Let $R$ be a real closed field, $S\subset R^k$ be a semi-algebraic set and consider the rational (co-)homology groups of $S$. It is a fundamental problem in computational real algebraic geometry to compute the dimensions of these rational vector spaces. We consider the special case, when the semi-algebraic set is defined by symmetric polynomials of fixed degree. The action of the symmetric group $S^k$ on $R^k$ gives these groups then the structure of a $S_k$-module. We study the associated isotypic decomposition and show bounds on the multiplicities of the irreducible representation appearing in this decomposition. In particular, we study the trivial representation, which is naturally isomorphic to the equivariant Homology groups, and given an algorithm with polynomially bounded (in $k$) complexity for computing these equivariant Betti numbers. We then discuss how this algorithm can be extended to an algorithm to compute the (ordinary) Betti numbers of S.

2010 Mathematics Subject Classification: 14P05
Key Words and Phrases: symmetric semi algebraic sets

⋅ 5th-D-09:30 − 10:00 [Changed: October 6th, 14:30 -- 15:00] A mobile application that reconstructs a 3D bony shape from un-calibrated radiographs (Kibeom Youn, Moon Seok Park, Jehee Lee)

윤기범*((서울대)), 박문석((서울대)), 이제희((서울대))
Kibeom Youn*, Seoul National University, Moon Seok Park, Seoul National University, Jehee Lee, Seoul National University

Computed tomography (CT)provides benefits in accurate diagnosis of bony deformities. However, the potential adverse effect of radiation exposure has become a concern, especially in a pediatric population as children are more susceptible to the effects of radiation. To reduce radiation dose while maintaining the accuracy of diagnosis, some systems have been proposed to reconstruct 3D bony shapes from calibrated bi-planar X-ray images. However, there are some problems to apply those systems to the actual medical environment. We propose a mobile application that reconstructs the 3D shape from uncalibrated radiographs. The user was required to take photos of AP and LAT radiographs with a iPad and segment the silhouette of the femur. If the user labels a part of the bone through the touch interface, the remaining bone parts are automatically extracted by the graph-cut algorithm. Each radiographic image is automatically aligned using a iterative PNP algorithm to match the silhouette of the statistical shape model, and then the shape of the femur is optimized.

2010 Mathematics Subject Classification: 65D18
Key Words and Phrases: 3D reconstruction, camera calibration, computed tomography, statistical shape model

⋅ 6th-G-13:30 − 15:30   Chair: Chair: Donghoon Hyeon (Seoul National University)

⋅ 6th-G-13:30 − 14:00 Multiview varieties and reconstruction problems (Makoto Miura)

Makoto Miura((고등과학원))
Makoto Miura, KIAS

In this talk, we discuss the geometry of multiple cameras in general dimensions. We give an algebro-geometric description of reconstruction problems in computer vision, and reformulate the projective reconstruction theorem by Hartley and Schaffalitzky. The main part of this talk is based on a joint work with Atsushi Ito and Kazushi Ueda.

2010 Mathematics Subject Classification: 14E05, 68T45
Key Words and Phrases: computer vision, algebraic vision, multiview geometry, projective reconstruction, Hilbert scheme

⋅ 6th-G-14:00 − 14:30 A new approach to nonnegativity and polynomial optimization (Mareike Dressler)

Mareike Dressler, Goethe University Frankfurt

Deciding nonnegativity of real polynomials is a key question in real algebraic geometry with crucial importance in polynomial optimization. It is well-known that in general this problem is co-NP-hard, therefore one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. Since the 19th century, sums of squares (SOS) are a standard certificate for nonnegativity, which can be recognized using semidefinite programming. This approach, however, has some issues, especially in practice if the optimization problem has many variables or high degree. In this talk I will introduce sums of nonnegative circuit polynomials (SONC). SONC polynomials are certain sparse polynomials having a special structure in terms of their Newton polytopes and supports and serve as a nonnegativity certificate for real polynomials, which is independent of sums of squares. Moreover I will give an overview about polynomial optimization via SONC polynomials. Similar as SOS correspond to SDP, the new SONC certificates correspond to geometric programming and relative entropy programming. Based on a Positivstellensatz for SONC polynomials we establish a converging hierarchy of efficiently computable lower bounds for constrained optimization problems. The talk is based on joint work with Sadik Iliman and Timo de Wolff.

2010 Mathematics Subject Classification: 14P10, 90C25, 12D05, 52B20, 14M25
Key Words and Phrases: nonnegative polynomials, sums of squares, sums of nonnegative circuit polynomials, sparsity, certificate, simplex, Positivstellensatz, semidefinite programming, geometric programming, relative entropy

⋅ 6th-G-14:30 − 15:00 [CANCELLED] Unlabeled triangulation (Andre Wagner)

Andre Wagner, HELLA Mobile Vision

In multiview geometry, when correspondences among multiple views are unknown the image points can be understood as being unlabeled. This is a common problem in computer vision. We give a novel approach to handle such situations of unlabeled point configurations. For unlabeled image points we design an algorithm that solves the triangulation problem with unknown correspondences. Further we deduce algebraic relations between unlabeled image points.

2010 Mathematics Subject Classification: 51A05
Key Words and Phrases: triangulation, multiview geometry, multilinear algebra

⋅ 6th-G-14:30 − 15:00 [Changed: October 6th, 14:30 -- 15:00] A mobile application that reconstructs a 3D bony shape from un-calibrated radiographs (Kibeom Youn, Moon Seok Park, Jehee Lee)

윤기범*((서울대)), 박문석((서울대)), 이제희((서울대))
Kibeom Youn*, Seoul National University, Moon Seok Park, Seoul National University, Jehee Lee, Seoul National University

Computed tomography (CT)provides benefits in accurate diagnosis of bony deformities. However, the potential adverse effect of radiation exposure has become a concern, especially in a pediatric population as children are more susceptible to the effects of radiation. To reduce radiation dose while maintaining the accuracy of diagnosis, some systems have been proposed to reconstruct 3D bony shapes from calibrated bi-planar X-ray images. However, there are some problems to apply those systems to the actual medical environment. We propose a mobile application that reconstructs the 3D shape from uncalibrated radiographs. The user was required to take photos of AP and LAT radiographs with a iPad and segment the silhouette of the femur. If the user labels a part of the bone through the touch interface, the remaining bone parts are automatically extracted by the graph-cut algorithm. Each radiographic image is automatically aligned using a iterative PNP algorithm to match the silhouette of the statistical shape model, and then the shape of the femur is optimized.

2010 Mathematics Subject Classification: 65D18
Key Words and Phrases: 3D reconstruction, camera calibration, computed tomography, statistical shape model

⋅ 6th-G-15:00 − 15:30 Bregman-divergence for Legendre exponential families and computer vision applications (Hyenkyun Woo)

우현균((한국기술교육대))
Hyenkyun Woo, Korea University of Technology and Education

Bregman-divergence is a well-known generalized distance framework in various applications, such as machine learning and computer vision. In this talk, by using dual structure of the Bregman-divergence associated with the subclass of convex function of Legendre function, we analyze the structure of the Legendre exponential families whose cumulant function corresponds to the conjugate convex function of Legendre type. Actually, Legendre exponential families are the extended version of the regular exponential families to include non-regular exponential families, such as the inverse Gaussian distribution. The main advantage of the proposed Bregman-divergence-based approach is that it offers systematic successive approximation tools to handle closed domain issues arising in non-regular exponential families and the statistical distributions having discrete random variables, such as Bernoulli distribution and Poisson distribution. In the end, as an application for computer vision, we introduce the generalized center-based clustering algorithm based on the extended Tweedie distribution.

2010 Mathematics Subject Classification: 62H30, 68T45, 14Q99
Key Words and Phrases: Bregman-divergence, Exponential families, clustering, convex function of Legendre type, machine learning

⋅ 6th-H-16:00 − 18:00   Chair: Kang Jin Han (DIGIST)

⋅ 6th-H-16:00 − 16:30 Deep convolutional framelets: A general deep learning framework for inverse problems (Jong Chul Ye, Yoseob Han, Eunju Cha)

예종철*((한국과학기술원)), 한요섭((한국과학기술원)), 차은주((한국과학기술원))
Jong Chul Ye*, KAIST, Yoseob Han, KAIST, Eunju Cha, KAIST

Recently, deep learning approaches with various network architectures have achieved significant performance improvement over existing iterative reconstruction methods in various imaging problems. However, it is still unclear {\em why} these deep learning architectures work for specific inverse problems. Moreover, in contrast to the usual evolution of signal processing theory around the classical theories, the link between deep learning and the classical signal processing approaches such as wavelets, non-local processing, compressed sensing, etc, are not yet well understood. To address these issues, here we show that the long-searched-for missing link is the convolution framelets for representing a signal by convolving local and non-local bases. The convolution framelets was originally developed to generalize the theory of low-rank Hankel matrix approaches for inverse problems, and this paper further extends the idea so that we can obtain a deep neural network using multilayer convolution framelets with perfect reconstruction (PR) under rectilinear linear unit nonlinearity (ReLU). Our analysis also shows that the popular deep network components such as residual block, redundant filter channels, and concatenated ReLU (CReLU) do indeed help to achieve the PR, while the pooling and unpooling layers should be augmented with high-pass branches to meet the PR condition. Moreover, by changing the number of filter channels and bias, we can control the shrinkage behaviors of the neural network. This discovery reveals the limitations of many existing deep learning architectures for inverse problems, and leads us to propose a novel theory for {\em deep convolutional framelets} neural network. Using numerical experiments with various inverse problems, we demonstrated that our deep convolution framelets network shows consistent improvement over existing deep architectures. This discovery suggests that the success of deep learning is not from a magical power of a black-box, but rather comes from the power of a novel signal representation using non-local basis combined with data-driven local basis, which is indeed a natural extension of classical signal processing theory.

2010 Mathematics Subject Classification: Primary 94A08, 97R40, 94A12, 92C55, 65T60, 42C40; Secondary 44A12
Key Words and Phrases: convolutional neural network, framelets, deep learning, inverse problems, ReLU, perfect reconstruction condition

⋅ 6th-H-16:30 − 17:00 Changing views on curves and surfaces (Kathl\'en Kohn, Bernd Sturmfels, Matthew Trager)

Kathl\'en Kohn*, Technical University of Berlin, Bernd Sturmfels, MPI MIS Leipzig \& University of California-Berkeley, Matthew Trager, Inria, ENS Paris, CNRS, PSL Research University

Visual events in computer vision are studied from the perspective of algebraic geometry. Given a sufficiently general curve or surface in 3-space, we consider the image or contour curve that arises by projecting from a viewpoint. Qualitative changes in that curve occur when the viewpoint crosses the visual event surface. We examine the components of this ruled surface, observe that these coincide with the iterated singular loci of the coisotropic hypersurfaces associated with the original curve or surface, and show how to compute exact representations for all visual event surfaces using algebraic methods.

2010 Mathematics Subject Classification: 14Q05, 14Q10, 14N10
Key Words and Phrases: visual event, image curve, singularities, coisotropic hypersurfaces

⋅ 6th-H-17:00 − 17:30 Mustafin varieties, tropical geometry and moduli spaces (Marvin Anas Hahn)

Marvin Anas Hahn, University of Tuebingen

Mustafin varieties are flat degenerations of projective spaces induced by a choice of an $n$-tuple of lattices in a vector space over a non-archimedean field. These varieties have a rich combinatorial structure as can be seen in pioneering work of Cartwright, H\"abich, Sturmfels and Werner. In this talk, we give a new description for Mustafin varieties in terms of images of rational maps, which were studied by Li. We completely classify the irreducible components of Mustafin varieties associated to a tuple of lattices. We relate Mustafin varieties for convex point configurations to Shimura varieties and certain moduli functors which appear in the limit linear series theory developed by Osserman called prelinked Grassmannians. Our methods include tropical convex hull computations and tropical intersection theory. This is talk is based on joint work with Binglin Li.

2010 Mathematics Subject Classification: 14T05, 14D06, 14D20
Key Words and Phrases: degenerations of projective spaces, tropical convexity, linked Grassmannians, tropical intersection theory, tropical linear spaces

⋅ 6th-H-17:30 − 18:00 Statistical learning under group actions, with applications to cryo-electron microscopy (Afonso Bandeira, Ben Blum-Smith, Joe Kileel, Amelia Perry, Jonathan Weed, Alexander Wein)

Afonso Bandeira, New York University, Ben Blum-Smith, New York University, Joe Kileel*, Princeton University, Amelia Perry, Massachusetts Institute of Technology, Jonathan Weed, Massachusetts Institute of Technology, Alexander Wein, New York University

Cryo-electron microscopy (cryo-EM) is an imaging technique that is currently revolutionizing structural biology; the Nobel Prize in Chemistry 2017 was awarded to Joachim Frank, Richard Henderson and Jacques Dubochet “for developing cryo-electron micro\-scopy for the high-resolution structure determination of biomolecules in solution”. Following cryo-EM experiments, one receives many very noisy 2D images of varying conformations of a 3D molecule, each taken from an unknown viewing direction. The noise, conformational variability, unknown orientations and big size of data are all aspects of cryo-EM which make reconstructing the 3D structure from the 2D images a formidable task. In this talk, I will place cryo-EM reconstruction inside a mathematical framework for statistical learning under group actions. The framework is rather general, and also applies to problems from signal processing, computer vision and computer graphics. The main result is the determination of the sample complexity for statistical learning under group actions, in high generality, in terms of the invariant theory of the underlying symmetry group. For cryo-EM, we contribute a novel ab initio 3D reconstruction algorithm which is sample-efficient and takes a single-pass through image data. This talk is based on joint work with Afonso Bandeira, Ben Blum-Smith, Amelia Perry, Jonathan Weed and Alexander Wein.

2010 Mathematics Subject Classification: 62F10, 92C55, 16W22, 15A72
Key Words and Phrases: statistical estimation, group action, cryo-EM, invariant theory, method of moments, tensor decomposition
Aperiodic Order and Dynamics

⋅ 4th-A-09:00 − 10:00   Chair: Michael Baake (Bielefeld University)

⋅ 4th-A-09:00 − 09:30 Scaling behaviour of diffraction intensities (Michael Baake, Uwe Grimm)

Michael Baake, University of Bielefeld, Uwe Grimm*, The Open University

The scaling behaviour of diffraction intensities near the origin has been used to classify the order of a structure, in particular with respect to the notion of hyperuniformity. In our talk, we address the scaling with some of the known techniques and results from rigorous diffraction theory. In particular, we consider substitution or inflation-based systems, for which we can employ renormalisation-type arguments, and link the scaling behaviour to Lyapunov exponents of a matrix iteration.

2010 Mathematics Subject Classification: 37B10 52C23 42A38
Key Words and Phrases: symbolic dynamics, aperiodic tilings, diffraction, scaling, Lyapunov exponent

⋅ 4th-A-09:30 − 10:00 Absence of absolutely continuous diffraction spectrum in inflation tilings (Franz G\"ahler)

Franz G\"ahler, Bielefeld University

The diffraction spectrum of an inflation tiling is given by the Fourier transform of its pair correlations, for which exact renormalisation equations have been developed recently. After reviewing these, we concentrate on the absolutely continuous (ac) part of the spectrum, which must satisfy the renormalisation equations separately. This allows us to derive necessary conditions for the existence of ac spectrum, and in fact to rule it out in many cases. We illustrate these techniques with a number of examples in one and two dimensions, where such a result was not previously known.

2010 Mathematics Subject Classification: 52C23, 42A38, 37D25
Key Words and Phrases: inflation rules, correlations, renormalisation, diffraction spectra, Lyapunov exponents

⋅ 4th-B-13:30 − 15:30   Chair: Jeong-Yup Lee (Catholic Kwandong University)

⋅ 4th-B-13:30 − 14:00 Statistics of patterns in typical cut and project sets (Alan Haynes, Antoine Julien, Henna Koivusalo, James Walton)

Alan Haynes, University of Houston, Antoine Julien*, Nord University, Henna Koivusalo, University of Vienna, James Walton, University of Glasgow

An important family of mathematical quasicrystals is generated by the ``cut and project" method. The principle is to project on a lower-dimensional plane a ``slice" of a higher-dimensional lattice. The properties of the resulting point-set depend crucially on the slope of the lower-dimensional plane. I will present some results regarding the repetition properties of a point-set obtained by using a ``typical" slope, in a measure-theoretic sense. This results extends some old results by Morse and Hedlund related to Sturmian sequences, and are based on techniques of Diophantine approximation.

2010 Mathematics Subject Classification: 52C23, 11K60
Key Words and Phrases: quasicrystals, diophantine, approximation recurrence

⋅ 4th-B-14:00 − 14:30 The Lagrange and Markov Spectra of Pythagorean triples (Byungchul Cha, Dong Han Kim)

차병철((Muhlenberg College)), 김동한*((동국대))
Byungchul Cha, Muhlenberg College, Dong Han Kim*, Dongguk University

Call $(p, q)$ a Pythagorean pair if $p$ and $q$ are positive coprime integers such that $p^2 + q^2$ is a perfect square. Draw a line $\ell$ from the origin into the first quadrant of the $xy$-plane. Suppose we want $\ell$ to avoid all but finitely many Pythagorean pairs with as large a margin as possible. Which $\ell$ is best? Which is second best? This is Diophantine approximation for the rational points in the unit circle. We will introduce a dynamical system originally defined by Romik in 2008, study its Lagrange and Markov spectra, and explain how our results can answer the above questions. This will provide a classical counterpart to our theory on Romik's dynamical system.

2010 Mathematics Subject Classification: 11J06, 11J70
Key Words and Phrases: Lagrange spectrum, Markov spectrum, Diophantine approximation

⋅ 4th-B-14:30 − 15:00 The unitary almost-Mathieu operator (Jake Fillman, Darren C. Ong, Zhenghe Zhang)

Jake Fillman, Virginia Tech, Darren C. Ong*, Xiamen University Malaysia, Zhenghe Zhang, University of California-Riverside

We discuss spectral characteristics of a one-dimensional quantum walk whose coins are distributed quasi-periodically. The unitary update rule of this quantum walk shares many spectral characteristics with the critical Almost-Mathieu Operator; however, it possesses a feature not present in the Almost-Mathieu Operator, namely singularity of the associated cocycles (this feature is, however, present in the so-called Extended Harper's Model). We show that this operator has empty absolutely continuous spectrum and that the Lyapunov exponent vanishes on the spectrum; hence, this model exhibits Cantor spectrum of zero Lebesgue measure for all irrational frequencies and arbitrary phase, which in physics is known as Hofstadter's butterfly. In fact, we will show something stronger, namely, that all spectral parameters in the spectrum are of critical type, in the language of Avila's global theory of analytic quasiperiodic cocycles. We further prove that it has empty point spectrum for each irrational frequency and away from a frequency-dependent set of phases having Lebesgue measure zero. The key ingredients in our proofs are an adaptation of Avila's Global Theory to the present setting, self-duality via the Fourier transform, and a Johnson-type theorem for singular dynamically defined CMV matrices which characterizes their spectra as the set of spectral parameters at which the associated cocycles fail to admit a dominated splitting.

2010 Mathematics Subject Classification: 47A10, 81Q12
Key Words and Phrases: unitary operators, almost-Mathieu operator, cocycles, spectral theory, quantum walks

⋅ 4th-B-15:00 − 15:30 Mean almost periodicity and pure point diffraction (Timo Spindeler, Nicolae Strungaru)

Timo Spindeler*, University of Alberta, Nicolae Strungaru, University of Alberta

The diffraction measure is a useful tool which can be used to determine the order (or disorder) of a point set. Very ordered sets lead to a pure point diffraction measure. Since it is generally not easy to compute this measure, it is desirable to find equivalent characterisations. In this talk, we will discuss mean almost periodicity and its relation to pure point diffractive sets.

2010 Mathematics Subject Classification: 52C23, 43A25, 43A60
Key Words and Phrases: Mean almost periodicity, pure point diffraction

⋅ 5th-E-13:30 − 15:00   Chair: Uwe Grimm (The Open University)

⋅ 5th-E-13:30 − 14:00 Linear repetitivity for polytopal cut and project sets (Jamie Walton, Henna Koivusalo)

Jamie Walton*, University of Glasgow, Henna Koivusalo, University of Vienna

I will discuss recent work with Henna Koivusalo that concerns the question of which Euclidean cut and project schemes with polytopal windows lead to patterns that are linearly repetitive. This builds upon previous work with Haynes and Koivusalo resolving the question for so-called `cubical windows'. We find a necessary and sufficient condition for linear repetitivity which applies to a large class of schemes satisfying a minor natural strengthening of the standard `almost canonical' property. The condition, and the proof of equivalence with linear repetitivity, involves an interesting blend of discrete geometry and Diophantine approximation.

2010 Mathematics Subject Classification: 52C23 11K60
Key Words and Phrases: cut and project sets, linear repetitivity, discrete geometry, lattices, Diophantine approximation

⋅ 5th-E-14:00 − 14:30 Toeplitz subshifts with entropy dimension (Uijin Jung)

정의진((아주대))
Uijin Jung, Ajou University

An infinite sequence $x$ is called a Toeplitz sequence if each word in $x$ occurs periodically in $x$. A subshift is called Toeplitz if it is an orbit closure of a Toeplitz sequence. In 80's Williams showed that any nonnegative real number can be realized as a topological entropy of a Toeplitz subshift. If a system has entropy 0, then the notion of entropy dimension can be used to capture the subexponential complexity of zero entropy systems. Given a real number $\alpha \in [0,1]$, we construct regular and nonregular Toeplitz subshifts with entropy dimension $\alpha$. Furthermore, for each positive integer $n$, we present a nonregular Toeplitz subshift with entropy dimension $\alpha$ which has at least $n$ ergodic measures. This is a joint work with Jungseob Lee, Kyewon Koh Park, and Jisang Yoo.

2010 Mathematics Subject Classification: 37B10, 37B40
Key Words and Phrases: Toeplitz, shift space, entropy dimension

⋅ 5th-E-14:30 − 15:00 Hausdorff dimension and entropy (Seonhee Lim)

임선희((서울대))
Seonhee Lim, Seoul National University

We will discuss the relation between Hausdorff dimension of badly approximable vectors in inhomogeneous Diophantine approximation and the entropy of a certain flow on a homogeneous space. We will also explain that maximality of such Hausdorff dimension implies invariance of the set under a certain subgroup of the (affine) special linear group. This is partly a joint work with U. Shapira and N. de Saxce, and partly a joint work with Y. Bugeaud, D. Kim, and M. Rams.

2010 Mathematics Subject Classification: 37D40, 37D20, 11J13
Key Words and Phrases: dynamics of diagonal flow, Diophantine approximation, Hausdorff dimension, entropy
Stochastic Processes and related Fields

⋅ 4th-A-09:00 − 10:00   Chair: Gerald Trutnau (Seoul National University)

⋅ 4th-A-09:00 − 09:30 Stochastic motion of droplets in the Cahn-Hilliard equation (Dirk Bl\"omker, Alexander Schindler)

Dirk Bl\"omker*, University of Augsburg, Alexander Schindler, University of Augsburg

We study the stochastic Cahn-Hilliard equation, which describes the phase separation and subsequent coarsening of binary alloys. In the nucleation regime almost spherical droplets appear, and we approximate the infinite dimensional stochastic dynamics of these droplets by the motion along a finite dimensional slow manifold. The main results are effective equations (given by stochastic ordinary differential equations) on the slow manifold and the stochastic stability of the manifold.

2010 Mathematics Subject Classification: 60H15
Key Words and Phrases: stochastic slow manifold, Cahn-Hilliard

⋅ 4th-A-09:30 − 10:00 Stochastic quantization for a fractional polymer measure (Wolfgang Bock, Torben Fattler, Jos\'e Luís da Silva, Ludwig Streit)

Wolfgang Bock, University of Kaiserslautern, Torben Fattler*, University of Kaiserslautern, Jos\'e Luís da Silva, CIMA, University of Madeira, Ludwig Streit, Bielefeld University

We prove existence of a diffusion process whose invariant measure is the fractional polymer or Edwards measure for fractional Brownian motion in dimension $d\in\mathbb{N}$ with Hurst parameter $H\in(0,1)$ fulfilling $Hd\leq 1$. The diffusion is constructed via Dirichlet form techniques in infinite dimensional (Gaussian) analysis. In the case $Hd<1$ by providing a Fukushima decomposition for the stochastic quantization of the fractional Edwards measure we show that the constructed process solves weakly a stochastic differential equation in infinite dimension for quasi–all starting points. Moreover, the solution process is driven by an Ornstein–Uhlenbeck process taking values in an infinite dimensional space and is unique, in the sense that the underlying Dirichlet form is Markov unique. The equilibrium measure, which is by construction the fractional Edwards measure, is specified to be an extremal Gibbs state.

2010 Mathematics Subject Classification: 81S20, 60G15, 60G22
Key Words and Phrases: stochastic quantization, Gaussian processes, fractional Brownian motion

⋅ 4th-B-13:30 − 15:30   Chair: Minjung Gim (NIMS)

⋅ 4th-B-13:30 − 14:00 Approximation of the stochastic heat equation with sticky reflected boundary condition (Martin Grothaus, Torben Fattler, Robert Vosshall)

Martin Grothaus*, University of Kaiserslautern, Torben Fattler, University of Kaiserslautern, Robert Vosshall, University of Kaiserslautern

In this talk we study the stochastic heat equation with sticky reflected boundary condition. Dirichlet form techniques are used to construct its solution. The obtained process already for some time is conjectured to be the scaling limit of the dynamical wetting model, also known as Ginzburg-Landau dynamics with pinning and reflection competing on the boundary. In this talk it is planned to discuss the recent progress on this problem.

2010 Mathematics Subject Classification: 60K35, 60J50, 60J55, 60B12, 60H15
Key Words and Phrases: interacting sticky reflected distorted Brownian motion, Skorokhod decomposition, interface models, scaling limit, SPDEs

⋅ 4th-B-14:00 − 14:30 Nonlinear Fokker-Planck-Kolmogorov equations and stochastic distribution dependent SDE (Viorel Barbu, Michael R\"ockner)

Viorel Barbu, Romanian Academy, Michael R\"ockner*, Bielefeld University and AMSS \& Chinese Academy of Sciences, Beijing

By Ito's formula the time marginals of a solution to a distribution dependent SDE solve a nonlinear Fokker-Planck-Kolmogorov equation. This talk is about the converse: we present a general technique how to identify a solution to a nonlinear Fokker-Planck-Kolmogorov equation consisting of probability densities as the time marginals of a solution to a distribution dependent SDE. We apply this to the special case of a porous media equation perturbed by the divergence of a vector field depending nonlinearly on the solution. More precisely, we construct a generalized entropic solution $u$ to this equation and apply the above general technique to find the corresponding distribution dependent SDE which has a weak solution with marginals given by $u$. We thus gain a probabilistic representation of $u$. The final aim is to develop a general theory relating distribution dependent SDE and nonlinear Fokker-Planck-Kolmogorov equations analogous to the classical linear case. \begin{thebibliography}{1} \bibitem{1} V. Barbu and M. R\"ockner, {\it Probabilistic representation of solutions to Fokker-Planck equations}, SIAM J. Math. Anal. {\bf 50} (2018), no. 4, 4246--4260 and arXiv:1801.10510. \end{thebibliography}

2010 Mathematics Subject Classification: 60H30, 60H10, 60G46, 35C99, 58J65
Key Words and Phrases: nonlinear FPKE, distribution dependent SDE

⋅ 4th-B-14:30 − 15:00 On the geometry of very rough Weierstrass curves whose components are not controlled (Peter Imkeller, Goncalo dos Reis, Olivier Pamen)

Peter Imkeller*, Humboldt University of Berlin, Goncalo dos Reis, University of Edinburgh, Olivier Pamen, AIMS Ghana

We investigate geometric properties of Weierstrass curves with two components, representing series based on trigonometric functions. They are seen to be $\frac{1}{2}-$ H\"older continuous, and are not (para-)controlled with respect to each other in the sense of the recently established Fourier analytic approach of rough path analysis. Their graph is represented as an attractor of a smooth random dynamical system. Our argument that its graph has Hausdorff dimension 2 is in the spirit of Ledrappier-Young's approach of the Hausdorff dimension of attractors. For one dimensional versions of the curves we establish duality between smoothness of the Sinai-Bowen-Ruelle measure of the dynamical system and the existence of a local time. This is joint work with G. dos Reis (U Edinburgh) and O. Pamen (U Liverpool and AIMS Ghana).

2010 Mathematics Subject Classification: 60H15, 37H05, 42A55
Key Words and Phrases: Weierstrass function, dynamical system, Hausdorff dimension, local time, Sinai-Bowen-Ruelle measure

⋅ 4th-B-15:00 − 15:30 Generalized couplings (Michael Scheutzow)

Michael Scheutzow, Technical University of Berlin

We introduce the concept of a generalized coupling and show how it can be used to show weak uniqueness and ergodicity of stochastic differential equations (with or without delay). This is joint work with Oleg Butkovsky and Alexey Kulik.

2010 Mathematics Subject Classification: 60H10
Key Words and Phrases: coupling, ergodicity, weak solution of an sde

⋅ 5th-E-13:30 − 15:30   Chair: Wilhelm Stannat (Technical University of Berlin)

⋅ 5th-E-13:30 − 14:00 Isomorphism theorems for Ginzburg-Landau fields (Jean Dominique Deuschel)

Jean Dominique Deuschel, Technical University of Berlin

We derive certain identities in law relating functionals of convex gradient fields to the local times of corresponding random walks in the associated Helffer-Sj\"ostrand representation. When restricting these identities to Gaussian measures, one recovers classical isomorphism theorems due to Dynkin, Ray-Knight and Le Jan. We apply these results to prove the existence of mass gaps for a class of anharmonic models with suitable single-spin distribution, thus extending results of Brydges, Fr\"ohlich and Spencer. This is a joint work with P.-F. Rodriguez.

2010 Mathematics Subject Classification: 60J27
Key Words and Phrases: Gaussian measures anharmonic model local time

⋅ 5th-E-14:00 − 14:30 Levy-Khintchine random matrices (Paul Jung)

폴정((한국과학기술원))
Paul Jung, KAIST

We study a class of Hermitian random matrices which includes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as adjacency matrices of Erd\H{o}s-R\'enyi random graphs with $p_n\sim\frac 1 n$. Our $n\times n$ random matrices have real entries which are i.i.d. up to symmetry. The distribution of entries depends on $n$, and we require row sums to converge in distribution; it is then well-known that the limit distribution must be infinitely divisible. We show that a limiting empirical spectral distribution (LSD) exists, and via local weak convergence of associated graphs, the LSD corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges called ``cords to infinity''. One example covered by the results are matrices with i.i.d. entries having infinite second moments, but normalized to be in the Gaussian domain of attraction. In this case, the limiting graph is $\mathbb{N}$ rooted at 1, so the LSD is the semi-circle law.

2010 Mathematics Subject Classification: 15B52, 60B20, 60G51
Key Words and Phrases: empirical spectral distribution, Wigner matrices, heavy-tailed random matrices, sparse random matrices, Erd\H{o}s-R\'enyi graph, local weak convergence

⋅ 5th-E-14:30 − 15:00 Stochastic heat diffusion near boundary (Kijung Lee)

이기정((아주대))
Kijung Lee, Ajou University

In this talk we discuss the heat diffusion under stochastic disturbance in a domain with boundary. The zero temperature control on the boundary forces the temperature to change in steep manner. This change near boundary is steep under deterministic disturbance and it gets much steeper under stochastic disturbance. In particular, the derivatives of the temperature near the boundary may blow up. To describe this behavior we are to use appropriate weights near the boundary. As the behavior depends on the shape of boundary, the weight has to be customized accordingly. We discuss this with half space domains and infinite wedge domains in $\mathbb{R}^2$.

2010 Mathematics Subject Classification: 60H15
Key Words and Phrases: stochastic heat diffusion, boundary behavior

⋅ 5th-E-15:00 − 15:30 Wasserstein diffusion and Dean-Kawasaki dynamics in 1D: Particle models for an ill posed SPDE (Max von Renesse)

Max von Renesse, University of Leipzig

Glassy materials in physics are often described by a class of SPDE with a particular multiplicative noise structure also known from macroscopic fluctuation theory. The easiest example is the so-called Dean-Kawasaki equations for super cooled liquids. We show that this SPDE is fundamentally ill-posed in the interesting parameter regimes and propose a correction which allows for solutions in terms of zero-range interacting particle systems which undergo a continuous branching and coagulation mechanism. Moreover, we show that this process realizes a Wasserstein diffusion in the sense of an associated Varadhan formula for the short time asymptotics governed by the quadratic Wasserstein metric of optimal transportation. Joint work with Vitalii Konarovkskyi and Tobias Lehmann (U. Leipzig).

2010 Mathematics Subject Classification: 60J45, 60H15, 60J50
Key Words and Phrases: Varadhan formula, Wasserstein metric, Dean-Kawasaki equation
Regularity theory of nonlinear elliptic, parabolic and variational problems

⋅ 4th-A-09:00 − 10:00   Chair: Sun-Sig Byun (Seoul National University)

⋅ 4th-A-09:00 − 09:30 Regularity of discretely p-harmonic functions on fully adaptive meshes (Lars Diening, Toni Scharle)

Lars Diening*, Bielefeld University, Toni Scharle, Oxford University

It is well known that harmonic functions and p-harmonic functions have higher interior regularity. In 1957 De Giorgi introduced a new technique that allows to obtain interior estimates of the form \begin{align*} \max_B |u|^p &\leq c\, \frac{1}{|2B|}\int_{2B} |u|^p\,dx. \end{align*} A similar result holds for $p$ harmonic functions. The proof is based on a subtle use of truncation operators and Cacciopoli estimates. In this talk we extend this technique and result to the setting of finite element solutions. Our solutions can be scalar valued as well as vector valued, which makes a big difference for~$p$-harmonic functions. Such estimates are of strong interest, since these estimates provide an alternative approach to the $L^\infty$-estimates of Schatz-Wahlbin for discretely harmonic functions (linear case). The main novelty of our approach is that we allow for fully adaptive meshes in contrast to uniform or slowly graded meshes.

2010 Mathematics Subject Classification: 35J70, 65N30
Key Words and Phrases: degenerate PDE, optimal regularity

⋅ 4th-A-09:30 − 10:00 Partial regularity for degenerate quasi-convex functionals with general growth and continuous coefficients (Pietro Celada, Jihoon Ok)

Pietro Celada((University of Parma)), 옥지훈*((경희대))
Pietro Celada, University of Parma, Jihoon Ok*, Kyung Hee University

We consider minimizers of degenerate quasi-convex functionals with continuous coefficients satisfying general growth conditions and prove the minimizers are H\"older continuous for all H\"older exponents $\alpha\in(0,1)$. The main tools are improved harmonic approximations in the Orlicz setting and Ekeland's variational principle.

2010 Mathematics Subject Classification: 49N60, 35J70, 26B25
Key Words and Phrases: partial regularity, quasi-convex functional, Oricz function

⋅ 4th-B-13:30 − 15:30   Chair: Lars Diening (Bielefeld University)

⋅ 4th-B-13:30 − 14:00 Regularity results for integrodifferential operators (Moritz Kassmann)

Moritz Kassmann, Bielefeld University

We present results on H\"older-regularity of solutions to integrodifferential equations driven by a nonlocal Dirichlet form. We include cases of anisotropic jump measures that are singular with respect to the underlying measure. The talk is based on recent results that were obtained together with Jamil Chaker and Bartlomiej Dyda.

2010 Mathematics Subject Classification: 35B65, 47G20
Key Words and Phrases: regularity, integro-differential operators

⋅ 4th-B-14:00 − 14:30 Global gradient estimates from composite materials (Yunsoo Jang)

장윤수((연세대))
Yunsoo Jang, Yonsei University

In this talk, we study global gradient estimates for weak solutions from composite materials. We assume that the domain is composed of a finite number of disjoint subdomains with Reifenberg flat boundaries and the coefficients are merely measurable in one variable and have small BMO semi-norms in the other variables on each subdomain. Our proof is based on a new observation for disjoint Reifenberg flat domains $\Omega^{k}$ and $\Omega^{l}$ that the normal vectors at $P \in \partial \Omega^{k}$ and $Q \in \partial \Omega^{l}$ are almost opposite if $P$ and $Q$ are close enough.

2010 Mathematics Subject Classification: 35J57, 74E30
Key Words and Phrases: composite material, gradient estimates, elliptic systems, Reifenberg flat domains, measurable coefficients

⋅ 4th-B-14:30 − 15:00 Entropy solutions of doubly nonlinear integro-differential equations (Petra Wittbold, Martin Scholtes)

Petra Wittbold*, University of Duisburg-Essen, Martin Scholtes, University of Duisburg-Essen

We consider a class of doubly nonlinear history-dependent diffusion equations. Our assumptions on the kernel include the case of a fractional derivative in the sense of Riemann-Liouville. Equations of this type have been proposed to model flows of fluids when memory effects arise. Existence and uniqueness of entropy solutions is established for general integrable data and Dirichlet boundary conditions. In the proof of existence the so-called fundamental identity is combind with a new regularization technique and classical monotonicity arguments.

2010 Mathematics Subject Classification: 45K05,47J35, 45D05, 35D99
Key Words and Phrases: integro-differential equations, dounbly nonlinear, entropy solution, integrable data

⋅ 4th-B-15:00 − 15:30 On Besov regularity of the $p$-Poisson equation in the plane (Anna Kh. Balci, Lars Diening, Markus Weimar)

Anna Kh. Balci*, Bielefeld University, Lars Diening, Bielefeld University, Markus Weimar, Ruhr-University Bochum

We consider $p$-Poisson equation with the right-hand side in divergence form: $$-{\rm div}(A(\nabla u))=-{\rm div} (|\nabla u|^{{p-2}}\nabla u)= -{\rm div} F. $$ It is known that the non-linear mapping $F\to A(\nabla u)$ satisfies the linear, optimal estimate $\|A(\nabla u)\|_X \le c\|F\|_X$ for several choices of spaces $X$ (Lebesgue spaces $L^q$ for $q\ge p'$, H\"older spaces $C^{0,\alpha}$ for some $\alpha\ge 0$, BMO). Our main result is the following Calder\'on-Zygmund-type regularity estimate in terms of the norm of the Besov Space $$\|A(\nabla u)\|_{B^s_{\rho,q}} \lesssim\|F\|_{B^s_{\rho,q}} + \text{lower order terms}.$$ The estimate is restricted to the case of the plane and $p\ge 2$.

2010 Mathematics Subject Classification: 35B65
Key Words and Phrases: $p$-Poisson equation, higher regularity

⋅ 5th-E-13:30 − 15:30   Chair: Sun-Sig Byun (Seoul National University)

⋅ 5th-E-13:30 − 14:00 Evolution equations in time-dependent domains (Christoph Scheven)

Christoph Scheven, University Duisburg-Essen

The talk deals with the Cauchy-Dirichlet problem for parabolic systems of $p$-Laplace type on non-cylindrical domains in space-time. In other words, we consider spatial domains whose shape varies in time. In the case of a growing domain, the boundary values can be interpreted as additional initial conditions, while in the case of a shrinking domain, the boundary values can be seen as a kind of obstacle condition. The treatment of time-varying domains turns out to be significantly harder than the standard case of cylindrical domains. We present an existence result for solutions to such problems under very weak regularity assumptions on the domain. A first regularity result for the constructed solutions guarantees that they depend continuously on time with respect to the $L^2$-norm if the domain does not shrink too fast.

2010 Mathematics Subject Classification: 35K40, 35A15, 35R05, 35A05
Key Words and Phrases: parabolic $p$-Laplace, variational solutions, noncylindrical domains, existence, continuity

⋅ 5th-E-14:00 − 14:30 Regularity for multi-phase variational problems (Cristiana De Filippis, Jehan Oh)

Cristiana De Filippis((Oxford University)), 오제한*((Bielefeld University))
Cristiana De Filippis, Oxford University, Jehan Oh*, Bielefeld University

We consider a multi-phase energy functional whose ellipticity rate and growth radically change according to variable coefficients. The aim of the talk is to show the H\"older regularity for the gradients of local minimizers of the multi-phase energy functional under sharp assumptions relating the growth couples to the H\"older exponents of the modulating coefficients. The talk is based on the joint work with Cristiana De Filippis.

2010 Mathematics Subject Classification: 49N60, 35J70
Key Words and Phrases: multi phase problem, regularity

⋅ 5th-E-14:30 − 15:00 Global regularity for quasilinear parabolic equations involving measure data (Sun-Sig Byun, Jung-Tae Park, Pilsoo Shin)

변순식((서울대)), 박정태*((고등과학원)), 신필수((서울대))
Sun-Sig Byun, Seoul National University, Jung-Tae Park*, KIAS, Pilsoo Shin, Seoul National University

In this talk we present global Calder\'on-Zygmund type estimates for parabolic $p$-Laplace type problems in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite mass. We define suitable solutions and provide minimal conditions which guarantee regularity results for such measure data problems.

2010 Mathematics Subject Classification: 35K92, 35B65
Key Words and Phrases: quasilinear parabolic equation, measure data, Calderon-Zygmund estimate, Reifenberg flat domain

⋅ 5th-E-15:00 − 15:30 The $p$-Laplace system with right-hand side in divergence form: Inner and up to the boundary Campanato estimates (Sebastian Schwarzacher)

Sebastian Schwarzacher, Charles University

In this talk we collect some very recent estimates for the gradient of solutions, and for the solutions themselves, to the $p$-Laplace system with right-hand side in divergence form. Both estimates inside the domain for local solutions, and global estimates for solutions to boundary value problems are discussed. Their formulation involves sharp maximal operators, whose properties enable us to translate some aspects of the elliptic regularity theory into a merely harmonic–analytic framework. As a consequence, a flexible, comprehensive approach to estimates for solutions to the $p$-Laplace system for a broad class of norms is derived. In particular, global estimates under minimal boundary regularity are presented. This is a joint work with: D. Breit, A. Cianchi, L. Diening and T. Kuusi.

2010 Mathematics Subject Classification: 49J40, 47J20, 47J40, 74H30
Key Words and Phrases: elliptic systems, gradient estimates, boundary regularity
Nonlinear Partial Differential Equations: Hyperbolic and mixed problems

⋅ 4th-A-09:00 − 10:00   Chair: Seung-Yeal Ha (Seoul National University)

⋅ 4th-A-09:00 − 09:30 Emergence of unstable modes for magnetohydrodynamic shock waves (Heinrich Freistuhler, Felix Kleber, Johannes Schropp)

Heinrich Freistuhler*, University of Konstanz, Felix Kleber, University of Konstanz, Johannes Schropp, University of Konstanz

This talk studies classical magnetohydrodynamic shock waves in an inviscid fluidic plasma that is assumed to be a perfect conductor of electricity. It identifies, partly numerically, partly analytically, critical manifolds in parameter space, across which slow or fast MHD shock waves undergo emergence of a complex conjugate pair of unstable transverse modes. For slow shocks, this emergence occurs in a particularly interesting way already in the parallel case, in which it happens at the spectral value $\hat\lambda\equiv\lambda/|\omega|=0$ and the critical manifold possesses an explicit algebraic representation. Within the set of non-parallel slow shocks the unstable mode pair emerges from \emph{two}\ generically different spectral values $\hat\lambda=\pm i\gamma$. For fast shocks, the critical manifold does not intersect the parallel regime, while the emergence within the set of non-parallel fast shocks again starts from two\ generically different spectral values.

2010 Mathematics Subject Classification: 35L67, 76E25, 76L05, 76W05
Key Words and Phrases: magnetohydrodynamics, shock waves, stability, continuation of invariant subspaces

⋅ 4th-A-09:30 − 10:00 Geometric dispersive equations on hyperbolic space (Andrew Lawrie, Jonas Luehrmann, Sung-Jin Oh, Sohrab Shahshahani)

Andrew Lawrie((Massachusetts Institute of Technology)), Jonas Luehrmann((Johns Hopkins University)), 오성진*((고등과학원)), Sohrab Shahshahani((University of Massachusetts at Amherst))
Andrew Lawrie, Massachusetts Institute of Technology, Jonas Luehrmann, Johns Hopkins University, Sung-Jin Oh*, KIAS, Sohrab Shahshahani, University of Massachusetts at Amherst

In this talk, I will report on some recent progress on the study of geometric dispersive equations on hyperbolic space. A particular emphasis will be on the rich family of finite energy harmonic maps from the hyperbolic plane, and their dynamic stability properties with respect to geometric dispersive equations.

2010 Mathematics Subject Classification: 35L70, 58J45
Key Words and Phrases: geometric wave equation, geometric dispersive equation, hyperbolic space

⋅ 4th-B-13:30 − 15:30   Chair: Robert Denk (University of Konstanz)

⋅ 4th-B-13:30 − 14:00 Blow up criteria for compressible Navier Stokes flow (Hi Jun Choe, Minsuk Yang)

최희준*((연세대)), 양민석((연세대))
Hi Jun Choe*, Yonsei university, Minsuk Yang, Yonsei university

We study the strong solution to the 3-D compressible Navier--Stokes equations. We propose a new blow up criterion for barotropic gases in terms of the integral norm of density $\rho$ and the divergence of the velocity $\mathbf u$ without any restriction on the physical viscosity constants. Our blow up criteria can be seen as a partial realization of the underlying principle that the higher integrability implies the boundedness and then eventual regularity. We also present similar blow up criterion for the heat conducting gases.

2010 Mathematics Subject Classification: 35-00
Key Words and Phrases: blow up, compressible Navier-Stokes flow, integral criterion, Beale-Kato-Majda style

⋅ 4th-B-14:00 − 14:30 [CANCELLED] Hyperbolic structures in diffuse-interface models for compressible two-phase flow (Christian Rohde)

Christian Rohde, University of Stuttgart

Mathematical models that describe the dynamics two-phase flows can be classified as either sharp-interface (SI) models or diffuse-interface (DI) models. For compressible liquid-vapour flow the first case results in a mixed-type elliptic-hyperbolic free boundary value problem. In the DI case one obtains higher-order systems like the class of Navier-Stokes-Korteweg equations which include e.g. third-order derivatives of the density. In both cases the mixed or complex type of the systems makes the analysis of the systems quite intricate, with many questions on wellposedness still being open. In the talk we propose a new hyperbolic-parabolic model approach to DI models that can be seen as an approximation of the original DI model. This approximation will be justified rigorously. One can apply standard analytical techniques to exploit the underlying hyperbolic structure. In particular we will derive a new homogenization limit that governs compressible two-phase flow in a porous medium.

2010 Mathematics Subject Classification: 35L65, 76T10
Key Words and Phrases: compressible two-phase flow, shock waves and phase boundaries, homogenization

⋅ 4th-B-14:30 − 15:00 Focusing energy-critical nonlinear fractional Schr\"odinger equations (Yonggeun Cho)

조용근((전북대))
Yonggeun Cho, Chonbuk National University

In this talk we consider a nonlinear fractional Schr\"odinger equation with non-local dispersion $|\nabla|^\alpha$ and focusing energy-critical Hartree type nonlinearity $[-(|x|^{-2\alpha}*|u|^2)u]$. We will first discuss a global well-posedness of radial case in energy space by adopting Kenig-Merle arguments when the initial energy and initial kinetic energy are less than those of ground state, respectively. For this purpose long time perturbation, profile decomposition and localized virial inequality are utilized. As an application of the localized virial inequality I will provide a time blowup for energy critical Hartree equations via commutator technique.

2010 Mathematics Subject Classification: M35Q55
Key Words and Phrases: global well-posedness, blowup, focusing energy-critical nonlinearity, Strichartz estimate, profile decomposition, virial argument, Sobolev inequality for radial functions

⋅ 4th-B-15:00 − 15:30 A variational time discretization for compressible Euler equations (Fabio Cavalletti, Marc Sedjro, Michael Westdickenberg)

Fabio Cavalletti, SISSA Trieste, Marc Sedjro, AIMS Tanzania, Michael Westdickenberg*, RWTH Aachen University

We introduce a variational time discretization for the multi-dimensional gas dynamics equations, in the spirit of minimizing movements for curves of maximal slope. Each timestep requires the minimization of a work functional plus internal energy, over the cone of monotone transport maps. We prove convergence to measure-valued solutions for the pressureless gas dynamics and the compressible Euler equations.

2010 Mathematics Subject Classification: 35L65, 49J40, 82C40
Key Words and Phrases: compressible gas dynamics, optimal transport

⋅ 5th-E-13:30 − 15:30   Chair: Hyeong-Ohk Bae (Ajou University)

⋅ 5th-E-13:30 − 14:00 Systems of conservation laws modelling phase mixtures and phase transitions (Gerald Warnecke)

Gerald Warnecke, Otto-von-Guericke University Magdeburg

The talk will present some recent results of the group in Magdeburg in connection with phase mixtures and phase transitions. These results include some modeling aspects as well as solutions to Riemann problems for systems of conservation laws that model mixtures of water and vapor. The mixture systems are weakly hyperbolic systems. When the terms modelling phase transitions between the liquid and vapor states are turned off, delta shocks can occur. They indicate that a phase transition should take place. Various examples are discussed.

2010 Mathematics Subject Classification: 35L45, 35D30, 76N15
Key Words and Phrases: mixture conservation laws, weakly hyperbolic systems, Riemann problem

⋅ 5th-E-14:00 − 14:30 Striated regularity of transport equations with nonlocal velocity (Hantaek Bae, James Kelliher)

배한택*((울산과학기술원)), James Kelliher((University of California-Riverside))
Hantaek Bae*, UNIST, James Kelliher, University of California-Riverside

We prove that striated regularity of solutions to the transport equations with nonlocal velocity field, such as the incompressible Euler equation and the aggregation equation with the Newtonian potential, are propagated over the time of solution.

2010 Mathematics Subject Classification: 35Q35, 35Q92, 39A01, 76B03
Key Words and Phrases: striated regularity, Euler equation, aggregation equation

⋅ 5th-E-14:30 − 15:00 Multiplication in anisotropic spaces and a fluid-structure-interaction model (J\"urgen Saal)

J\"urgen Saal, University of Duesseldorf

Quasilinear mixed order systems arise in countless applications in natural sciences and technology. Important representatives of this class of PDE are free boundary problems in fluid dynamics. Relying on the maximal regularity approach, not seldom intricate nonlinearities of quasilinear mixed order systems have to be estimated in anisotropic (in space and time) function spaces. In my talk I would like to present recently derived results on multiplication and analytic Nemytskii operators on scales of anisotropic function spaces. By these results the estimation of nonlinear terms is essentially reduced to verifying an elementary condition for the corresponding anisotropic Sobolev indices. In the second part of my talk it is demonstrated, how these results can be utilized in order to derive well-posedness for a fluid-structure-interaction model.

2010 Mathematics Subject Classification: 76T99
Key Words and Phrases: fluid-structure-interaction

⋅ 5th-E-15:00 − 15:30 Global existence of weak solutions for Navier-Stokes-BGK system (Young-Pil Choi)

최영필((인하대))
Young-Pil Choi, Inha University

Recently, the study on particle-fluid system is gathering a lot of attentions due to their applications, for example, in the study of sedimentation phenomena, fuel injector in engines, and compressibility of droplets of the spray, etc. In this talk, we discuss the global existence of weak solutions for the system consisting of the BGK model of Boltzmann equation and incompressible Navier-Stokes equations coupled through a drag forcing term.

2010 Mathematics Subject Classification: 35Q30, 35Q20, 35Q62
Key Words and Phrases: Vlasov equation, BGK model, incompressible Navier-Stokes equations, spray models, global existence of weak solutions
Chemotactic and Fluid Dynamics

⋅ 5th-D-09:00 − 10:00   Chair: Jihoon Lee (Chung-Ang University)

⋅ 5th-D-09:00 − 09:30 A logistic population model with starvation-driven dispersal under a free boundary (InKyung Ahn, Wonhyung Choi)

안인경*((고려대)), 최원형((고려대))
InKyung Ahn*, Korea University, Wonhyung Choi, Korea University

In many cases, the movement of species within a region depends on the availability of food and other resources necessary for its survival. Starvation driven diffusion (SDD) is a dispersal strategy that increases the motility of biological organisms in unfavorable environments, i.e., a species moves more frequently in search of food if resources are insufficient (Cho and Kim, 2013). In this study, the proposed model represents the dispersion of an invasive species undergoing SDD, where the free boundary represents the expanding front. We observe that the spreading-vanishing dichotomy, which holds in the linear dispersal model (Zhou and Xiao, 2013), also holds in the model undergoing SDD. We also provide the estimates for the spreading speed of the free boundary during the spreading process. Finally, our results are compared with the results of the linear dispersal model to investigate the advantages of this strategic dispersal with respect to survival in new environments.

2010 Mathematics Subject Classification: 35K20, 35K55, 35R35
Key Words and Phrases: free boundary, asymptotic behavior, starvation-driven diffusion, spreading-vanishing dichotomy, spreading speed

⋅ 5th-D-09:30 − 10:00 Uniqueness of solutions for Keller-Segel system of porous medium type coupled to fluid equations (Hantaek Bae, Kyungkeun Kang, Seick Kim)

배한택*((울산과학기술원)), 강경근((연세대)), 김세익((연세대))
Hantaek Bae*, UNIST, Kyungkeun Kang, Yonsei University, Seick Kim, Yonsei University

We prove the uniqueness of H\"older continuous weak solutions via duality argument and vanishing viscosity method for the Keller-Segel system of porous medium type equations coupled to the Stokes system in dimensions three. An important step is the estimate of the Green function of parabolic equations with lower order terms of variable coefficients.

2010 Mathematics Subject Classification: 35A02, 35K65, 35Q92
Key Words and Phrases: Keller-Segel equation, Porous media, H\"older regularity, uniqueness, duality argument, vanishing viscosity, Green matrix of parabolic equation

⋅ 6th-G-13:30 − 15:30   Chair: Angela Stevens (University of Muenster)

⋅ 6th-G-13:30 − 14:00 How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases (Michael Winkler)

Michael Winkler, Paderborn University

The classical parabolic-elliptic Keller-Segel system is considered under homogeneous Neumann boundary conditions in the ball $\Omega=B_R(0)\subset \mathbb R^n$. The main objective is to reveal that in the context of radially symmetric solutions, this problem exhibits an apparently novel type of critical mass phenomenon: It is shown, namely, that for any choice of $n\ge 2$ and $R>0$ there exists a positive number $m_c=m_c(n,R)$ with the following properties: \begin{itemize} \item Whenever $m>m_c$, for any nonconstant nonnegative radial initial data $u_0$ with $\int_\Omega u_0=m$ which are, in an appropriately defined sense, more concentrated than the associated spatially homogeneous equilibrium determined by $u\equiv \frac{m}{|\Omega|}$, the corresponding initial-value problem admits a solution blowing up in finite time; in particular, this implies that any nonconstant and radially nonincreasing initial data $u_0$ with $\int_\Omega u_0>m_c$ enforce blow-up. \item If $m<m_c$, however, then there exist infinitely many nonnegative radial functions $u_0$ which satisfy $\int_\Omega u_0=m$ and which are more concentrated than $u\equiv \frac{m}{|\Omega|}$, but which yet allow for global bounded solutions emanating from $u_0$. \end{itemize} In consequence, precisely at mass levels above $m_c$ the constant steady states of the system possess the extreme instability property of repelling arbitrary concentration-increasing perturbations in such a drastic sense that corresponding trajectories collapse in finite time.

2010 Mathematics Subject Classification: 35B40, 35B44, 35K65, 35B33, 92C17
Key Words and Phrases: chemotaxis, critical mass, finite-time blow-up

⋅ 6th-G-14:00 − 14:30 A bulk-surface model for active-transport-induced polarisation (Matthias R\"oger, Keith Anguige)

Matthias R\"oger*, Technical University of Dortmund, Keith Anguige, University of Freiburg

In biological cells a tight regulation of processes in the cell interior and on the cell membrane is essential for many functions of the cell. We consider the spontaneous polarization of cells by a feedback mechanism induced by active transport processes. A mathematical model in the form of coupled bulk-surface reaction-drift-diffusion system is introduced that has some similarity with classical chemotaxis models. We evaluate the well-posedness of the system.

2010 Mathematics Subject Classification: 35Q92, 35B44, 35K57
Key Words and Phrases: PDEs on surfaces, cell polarisation, active transport, chemotaxis, blow-up

⋅ 6th-G-14:30 − 15:00 Self-similar lifting and persistent touch-down point solutions in the thin-film equation (Hans Kn\"upfer, Carlota Cuesta, Juan Velazquez)

Hans Kn\"upfer*, Heidelberg University, Carlota Cuesta, University of the Basque Country, Juan Velazquez, University of Bonn

We discuss the appearance of self-similar blow-up solutions for thin-film equations with different mobility exponents. This is related to non-uniqueness phenomena for weak solution of the same equation. The proof is based on dynamical systems arguments.

2010 Mathematics Subject Classification: 35K65, 34A34, 76D27
Key Words and Phrases: self-similar solutions, thin-film equation, non-uniqueness

⋅ 6th-G-15:00 − 15:30 Regularity and temporal asymptotics of Keller-Segel models coupled to fluid equations (Kyung Keun Kang)

강경근((연세대))
Kyung Keun Kang, Yonsei University

We study chemotaxis equations coupled to the Navier-Stokes equations, which is a mathematical model describing the dynamics of oxygen, swimming bacteria (Bacillus subtilis), and viscous incompressible fluids. It is unknown for parameters of general chemotatic sensitivity and consumption rate in two and three dimensions, whether or not regular solutions exist globally in time or develop a singularity in a finite time for regular data. We discuss existence of regular solutions and temporal decays of solutions as well as their asymptotics, under a certain type of conditions of parameters and initial data, as time tends to infinity. This is the joint work with M. Chae, K. Lee and J. Lee.

2010 Mathematics Subject Classification: 35Q30, 35Q35, 76Dxx, 76Bxx
Key Words and Phrases: asymptotics, Chemotaxis-Navier-Stokes, global well-posedness, temporal decay

⋅ 6th-H-16:00 − 18:00   Chair: Kyung Keun Kang (Yonsei University)

⋅ 6th-H-16:00 − 16:30 Chemotaxis and aggregation equations (Martin Burger, Marco Di Francesco, Simone Fagioli, Angela Stevens)

Martin Burger, University of Muenster, Marco Di Francesco, University of L'Aquila, Simone Fagioli, University of L'Aquila, Angela Stevens*, University of Muenster

Macroscopic models for systems involving diffusion, short-range repulsion, and long-range attraction have been studied extensively in the last decades. In this paper we extend the analysis to a system for two species interacting with each other according to different inner- and intra-species attractions. Under suitable conditions on this self- and crosswise attraction an interesting effect can be observed, namely phase separation into neighboring regions, each of which contains only one of the species. We prove that the intersection of the support of the stationary solutions of the continuum model for the two species has zero Lebesgue measure, while the support of the sum of the two densities is a connected interval. Preliminary results indicate the existence of phase separation, i.e., spatial sorting of the different species. A detailed analysis is given in one spatial dimension. The existence and shape of segregated stationary solutions is shown via the Krein–Rutman theorem. Moreover, for small repulsion/nonlinear diffusion, also uniqueness of these stationary states is proved.

2010 Mathematics Subject Classification: 35K59
Key Words and Phrases: phase separation, spatial sorting, nonlinear diffusion, long-range attraction, stationary states

⋅ 6th-H-16:30 − 17:00 Mathematical analysis of some biomixing model with biological reactions (Myeongju Chae, Kyungkeun Kang, Jihoon Lee)

채명주((한경대)), 강경근((연세대)), 이지훈*((중앙대))
Myeongju Chae, Hankyung National University, Kyungkeun Kang, Yonsei University, Jihoon Lee*, Chung-Ang University

In this talk, we consider coral broadcast spawning models involving diffusion, advection, chemotaxis, and reactions with different egg and sperm densities. We prove the global-in-time existence of the regular solutions of the models as well as their temporal decays. Furthermore, we show that the total masses of egg and sperm density have positive lower bounds as time extends to infinity in three dimensions. Similar results are also obtained for the coral spawning models coupled with the incompressible fluid equations.

2010 Mathematics Subject Classification: 35Q30
Key Words and Phrases: chemotaxis, well-posedness, biomixing

⋅ 6th-H-17:00 − 17:30 On the vortex filament conjecture in ideal fluids (Robert L. Jerrard, Christian Seis)

Robert L. Jerrard, University of Toronto, Christian Seis*, University of Muenster

We study the evolution of vortex filaments in ideal fluids. A conjecture, dating back to da Rios in 1906, states that if the vorticity is initially concentrated around a closed curve, it remains concentrated for some time and the evolution of the curve is geometrically described by the binormal curvature flow. We focus on the second part of this conjecture and derive the binormal curvature flow under a weak vorticity concentration condition. Our proof relies on estimates for the underlying Hamiltonian structures.

2010 Mathematics Subject Classification: 76B47
Key Words and Phrases: Euler equation, vortex filaments, binormal curvature flow

⋅ 6th-H-17:30 − 18:00 Asymptotic analysis of Vlasov equations with nonlocal forces (Young-Pil Choi)

최영필((인하대))
Young-Pil Choi, Inha University

Collective coordinated motion of autonomous self-propelled agents with self-organiza\-tion into robust patterns appears in many applications ranging from animal herding to the emergence of common languages. Apart from its biological and evolutionary relevance, collective phenomena play a prominent role in many other scientific disciplines, such as robotics, control theory, economics and social sciences. In this talk, we will consider kinetic type models describing the collective behaviors and discuss a quantitative estimate of a large friction limit to a continuity equation with nonlocal velocity fields, called aggregation equations, by employing 2-Wasserstein distance.

2010 Mathematics Subject Classification: 35Q70, 35Q92, 35Q31
Key Words and Phrases: aggregation equations, pressureless Euler equations, Vlasov equation, hydrodynamic limit, large friction limit
Structured nonparametric and high-dimensional statistics

⋅ 4th-A-09:00 − 10:00   Chair: Hee-Seok Oh (Seoul National University)

⋅ 4th-A-09:00 − 09:30 Additive functional regression for densities as responses (Kyunghee Han, Hans-Georg Mueller, Byeong Uk Park)

한경희((University of California-Davis)), Hans-Georg Mueller((University of California-Davis)), 박병욱*((서울대))
Kyunghee Han, University of California-Davis, Hans-Georg Mueller, University of California-Davis, Byeong Uk Park*, Seoul National University

We propose and investigate additive density regression, a novel additive functional regression model for situations where the responses are random distributions that can be viewed as random densities and the predictors are vectors. Data in the form of samples of densities or distributions are increasingly encountered in statistical analysis and there is a need for flexible regression models that accommodate random densities as responses. Such models are of special interest for multivariate continuous predictors, where unrestricted nonparametric regression approaches are subject to the curse of dimensionality. Additive models can be expected to maintain one-dimensional rates of convergence while permitting a substantial degree of flexibility. This motivates the development of additive regression models for situations where multivariate continuous predictors are coupled with densities as responses. To overcome the problem that distributions do not form a vector space, we utilize a class of transformations that map densities to unrestricted square integrable functions and then deploy an additive functional regression model to fit the responses in the unrestricted space, finally transforming back to density space. We implement the proposed additive model with an extended version of smooth backfitting and establish the consistency of this approach, including rates of convergence. The proposed method is illustrated with an application to the distributions of baby names in the United States.

2010 Mathematics Subject Classification: 62G08
Key Words and Phrases: additive models, functional data analysis, random densities, smooth backfitting

⋅ 4th-A-09:30 − 10:00 Nonparametric density estimation for intentionally corrupted functional data (Alexander Meister, Aurore Delaigle)

Alexander Meister*, University of Rostock, Aurore Delaigle, University of Melbourne

We consider statistical models where functional data are artificially contaminated by independent Wiener processes in order to satisfy privacy constraints. We show that the corrupted observations have a Wiener density which determines the distribution of the original functional random variables, masked near the origin, uniquely, and we construct a nonparametric estimator of that density. We derive an upper bound for its mean integrated squared error which has a polynomial convergence rate, and we establish an asymptotic lower bound on the minimax convergence rates which is close to the rate attained by our estimator. Our estimator requires the choice of a basis and of two smoothing parameters. We propose data-driven ways of choosing them and prove that the asymptotic quality of our estimator is not significantly affected by the empirical parameter selection. We examine the numerical performance of our method via simulated examples.

2010 Mathematics Subject Classification: 62G07
Key Words and Phrases: classification, convergence rates, differential privacy, infinite-dimensional Gaussian mixtures, Wiener densities

⋅ 4th-B-13:30 − 15:30   Chair: Woncheol Jang (Seoul National University)

⋅ 4th-B-13:30 − 14:00 Geometrizing rates of convergence under differential privacy constraints (Angelika Rohde, Lukas Steinberger)

Angelika Rohde*, University of Freiburg, Lukas Steinberger, University of Freiburg

We study the problem of estimating a functional $\theta(\mathbb P)$ of an unknown probability distribution $\mathbb P \in\mathcal P$ in which the original iid sample $X_1,\dots, X_n$ is kept private even from the statistician via an $\alpha$-local differential privacy constraint. Let $\omega_{TV}$ denote the modulus of continuity of the functional $\theta$ over $\mathcal P$, with respect to total variation distance. For a large class of loss functions $l$, we prove that the privatized minimax risk is equivalent to $l(\omega_{TV}(n^{-1/2}))$ to within constants, under regularity conditions that are satisfied, in particular, if $\theta$ is linear and $\P$ is convex. Our results complement the theory developed by Donoho and Liu (1991) with the nowadays highly relevant case of privatized data. Somewhat surprisingly, the difficulty of the estimation problem in the private case is characterized by $\omega_{TV}$, whereas, it is characterized by the Hellinger modulus of continuity if the original data $X_1,\dots, X_n$ are available. We also provide a general recipe for constructing rate optimal privatization mechanisms and illustrate the general theory in numerous examples. Our theory allows to quantify the price to be paid for local differential privacy in a large class of estimation problems.

2010 Mathematics Subject Classification: 62G05, 62C20
Key Words and Phrases: local differential privacy, minimax estimation, moduli of continuity

⋅ 4th-B-14:00 − 14:30 Statistical inference on party positions from texts: Statistical modeling, boostrap and adjusting for time effects (Eun Ryung Lee, Carsten Jentsch, Enno Mammen)

이은령*((성균관대)), Carsten Jentsch((Technical University of Dortmund)), Enno Mammen((Heidelberg University))
Eun Ryung Lee*, Sungkyunkwan University, Carsten Jentsch, Technical University of Dortmund, Enno Mammen, Heidelberg University

One central task in comparative politics is to locate party positions in a certain political space. For this purpose, several empirical methods have been proposed using text as data sources. In general, the analysis of texts to extract information is a difficult task. Its data structure is very complex and political texts usually contain a large number of words such that a simultaneous analysis of word counts becomes challenging. In this paper, we consider Poisson models for each word count simultaneously and provide a statistical analysis suitable for political text data. In particular, we allow for multi-dimensional party positions and develop a data-driven way of determining the dimension of positions. Allowing for multi-dimensional political positions gives new insights in the evolution of party positions and helps our understanding of a political system. Additionally, we consider a novel model which allows the political lexicon to change over time and develop an estimation procedure based on LASSO and fused LASSO penalization techniques to address high-dimensionality via significant dimension reduction. The latter model extension gives more insights into the potentially changing use of words by left and right-wing parties over time. Furthermore, the procedure is capable to identify automatically words having a discriminating effect between party positions. To address the potential dependence structure of the word counts over time, we included integer-valued time series processes into our modeling approach and we implemented a suitable bootstrap method to construct confidence intervals for the model parameters. We apply our approach to party manifesto data from German parties over all seven federal elections after German reunification. The approach is simply implemented as it does not require any a priori information (from external source) nor expert knowledge to process the data. The data studies confirm that our procedure is robust, runs stable and leads to meaningful and interpretable results.

2010 Mathematics Subject Classification: 62P25, 62H12
Key Words and Phrases: party manifestos, term document matrices, high-dimensional data

⋅ 4th-B-14:30 − 15:00 Ill-posed estimation in high-dimensional models with instrumental variables (Christoph Breunig, Enno Mammen, Anna Simoni)

Christoph Breunig*, Humboldt University of Berlin, Enno Mammen, Heidelberg University, Anna Simoni, CREST, CNRS

This paper is concerned with inference about low-dimensional components of a high-dimensional parameter vector β which is identified through instrumental variables. We allow for eigenvalues of the expected outer product of included and excluded covariates, denoted by M, to shrink to zero as the sample size increases. We propose a novel estimator based on desparsification of an instrumental variable Lasso estimator, which is a regularized version of 2SLS with an additional correction term. This estimator converges to β at a rate depending on the mapping properties of M captured by a sparse link condition. Linear combinations of our estimator of β are shown to be asymptotically normally distributed. Based on consistent covariance estimation, our method allows for constructing confidence intervals and statistical tests for single or low-dimensional components of β. In Monte-Carlo simulations we analyze the finite sample behavior of our estimator.

2010 Mathematics Subject Classification: 62G05
Key Words and Phrases: instrumental variables, sparsity, central limit theorem, lasso, linear model, desparsification, ill-posed estimation problem

⋅ 4th-B-15:00 − 15:30 Adaptive discrete smoothing for (high-dimensional and nonlinear) panel data (Martin Spindler, Xi Chen, Victor Chernozhukov, Ye Luo)

Martin Spindler*, University of Hamburg, Xi Chen, New York University, Victor Chernozhukov, Massachusetts Institute of Technology, Ye Luo, University of Hong Kong

In this paper we develop a data-driven smoothing technique for non-linear panel data models. We allow for individual specific (non-linear) functions and estimation with econometric or machine learning methods by using weighted observations from other individuals. The weights are determined by a data-driven way and depend on the similarity between the corresponding functions and are measured based on initial estimates. The key feature of such a procedure is that it clusters individuals based on the distance / similarity between them, estimated in a first stage. Our estimation method can be combined with various statistical estimation procedures, in particular modern machine learning methods which are in particular fruitful in the high-dimensional case and with complex, heterogeneous data. The methods can also be applied for estimation of Random Coefficient models and for estimation of nonparametric functions with many categorical variables (\textquotedblleft cells\textquotedblright). The theoretical properties of the proposed estimator are derived which improve on many classical methods. We conduct a simulation study which shows that the prediction can be greatly improved by using our estimator. Finally, we analyze a big data set from didichuxing.com, a leading company in transportation industry, to analyze and predict the gap between supply and demand based on a large set of covariates. Our estimator clearly performs much better in out-of-sample prediction compared to existing non-linear panel data estimators.

2010 Mathematics Subject Classification: 62Hxx
Key Words and Phrases: nonparametric econometrics, nonlinear panel data, discrete smoothing, incidental parameter problem, clustering

⋅ 5th-E-13:30 − 15:30   Chair: Kyusang Yu (Konkuk University)

⋅ 5th-E-13:30 − 14:00 Dirty limit theorems on noneuclidean spaces (Stephan Huckemann)

Stephan Huckemann, University of Goettingen, Felix Bernstein Institute for Mathematical Statistics in the Biosciences

For inference on means of random vectors, the central limit theorem (CLT) is a central tool. Fr\'echet (1948) extended the notion of means to arbitrary metric spaces, as minimizers of expected squared distance. For such, under mild conditions, a strong law has been provided 1977 by Ziezold and, in case of manifolds and additional stronger conditions, a CLT has been derived by Bhattacharya and Paragenaru (2005). In a local chart, this CLT features a classical normal limiting distribution with a classical rate of inverse square root of sample size. If these additional stronger conditions are not satisfied, CLTs may still hold but feature different rates and different limit distributions. We give examples of such ``dirty" limit theorems featuring faster rates (stickiness) and slower rates (smeariness). The former may occur on NNC (nonnegative curvature) spaces, here the distribution around cut loci of means plays a central role, the latter on NPC (nonpositive curvature) spaces. Both effects may have serious practical applications.

2010 Mathematics Subject Classification: 60F05, 62H11
Key Words and Phrases: central limit theorems, data on spaces with geometric structure

⋅ 5th-E-14:00 − 14:30 Inference on distribution functions under measurement error (Karun Adusumilli, Taisuke Otsu, Yoon-Jae Whang)

Karun Adusumilli((University of Pennsylvania)), Taisuke Otsu((London School of Economics)), 황윤재*((서울대))
Karun Adusumilli, University of Pennsylvania, Taisuke Otsu, London School of Economics, Yoon-Jae Whang*, Seoul National University

This paper is concerned with inference on the cumulative distribution function (cdf) $F_{X^{*}}$ in the classical measurement error model $X=X^{*}+\epsilon$. We show validity of asymptotic and bootstrap approximations for the distribution of the deviation in the sup-norm between the deconvolution cdf estimator of Hall and Lahiri (2008) and $F_{X^{*}}$. We allow the density of $\epsilon$ to be ordinary or super smooth, or to be estimated by repeated measurements. Our approximation results are applicable to various contexts, such as confidence bands for $F_{X^{*}}$ and its quantiles, and for performing various cdf-based tests such as goodness-of-fit tests for parametric models of densities, two sample homogeneity tests, and tests for stochastic dominance. Simulation and real data examples illustrate satisfactory performance of the proposed methods.

2010 Mathematics Subject Classification: 62G86
Key Words and Phrases: measurement error, deconvolution, stochastic dominance

⋅ 5th-E-14:30 − 15:00 Large ball probability and inference for spectral projectors (Vladimir Spokoiny)

Vladimir Spokoiny, Weierstrass Institute \& Humboldt University of Berlin

Let $X_{1}, \ldots,X_{n} $ be i.i.d. sample in $\mathbb R^{p}$ with zero mean and the covariance matrix $- \Sigma $. We consider the problem of confidence estimation of the projectors on an eigenspace of $\Sigma -$. This paper offers two procedures: one is based on the resampling technique; the other one is Bayesian and uses Bayesian calculus from the conjugated Wishart prior. Accuracy of both methods is evaluated with sharp error bounds. The study heavily uses recent results on ``large ball probability” for Gaussian measures in a Hilbert space.

2010 Mathematics Subject Classification: 62G05, 62E17
Key Words and Phrases: Gaussian comparison, Gaussian anti-concentration inequalities, dimension free bounds, Schatten norm, high-dimensional inference, high-dimensional CLT

⋅ 5th-E-15:00 − 15:30 Optimal estimation of sparse high-dimensional additive models (Karl Gregory, Enno Mammen, Martin Wahl)

Karl Gregory, University of South Carolina, Enno Mammen*, Heidelberg University, Martin Wahl, Humboldt University of Berlin

In this talk we discuss the estimation of a nonparametric component $f_1$ of a nonparametric additive model $Y=f_1(X_1) + \cdots+ f_q(X_q) + \varepsilon$. We allow the number $q$ of additive components to grow to infinity and we make sparsity assumptions about the number of nonzero additive components. We compare this estimation problem with that of estimating $f_1$ in the oracle model $Z= f_1(X_1) + \varepsilon$, for which the additive components $f_2,\dots,f_q$ are known. We construct a two-step presmoothing-and-resmoothing estimator of $f_1$ in the additive model and state finite-sample bounds for the difference between our estimator and some smoothing estimators $\hat f_1^{\text{(oracle)}}$ in the oracle model which satisfy mild conditions. In an asymptotic setting these bounds can be used to show asymptotic equivalence of our estimator and the oracle estimators; the paper thus shows that, asymptotically, under strong enough sparsity conditions, knowledge of $f_2,\dots,f_q$ has no effect on estimation accuracy. Our first step is to estimate all of the components in the additive model with undersmoothing using a group-Lasso estimator. We then construct pseudo responses $\hat Y$ by evaluating a desparsified modification of our undersmoothed estimator of $f_1$ at the design points. In the second step the smoothing method of the oracle estimator $\hat f_1^{\text{(oracle)}}$ is applied to a nonparametric regression problem with ``responses'' $\hat Y$ and covariates $X_1$. Our mathematical exposition centers primarily on establishing properties of the presmoothing estimator. We also present simulation results demonstrating close-to-oracle performance of our estimator in practical applications. The main results of the paper are also important for understanding the behavior of the presmoothing estimator when the resmoothing step is omitted.

2010 Mathematics Subject Classification: 62G08, 62G20
Key Words and Phrases: nonparametric curve estimation, additive models, penalization, Lasso, variable selection, dimension reduction
Probability Theory and Applications

⋅ 5th-D-09:00 − 10:00   Chair: Rene Schilling (Technical University of Dresden)

⋅ 5th-D-09:00 − 09:30 Maximal regularity results for stochastic partial differential equations with non-local operators (Kyeong-Hun Kim)

김경훈((고려대))
Kyeong-Hun Kim, Korea University

Stochastic partial equations (SPDEs) arise when one models natural phenomena under random environments. Especially, SPDEs with time fractional derivatives can be used to describe random effects on transport of particles subject to sticking and trapping. Also, SPDEs with spacial non-local operators can be used to describe random phenomena with pure jumps. In this talk, I will present uniqueness and existence results and maximal regularity theory for stochastic partial differential equations with various non-local operators.

2010 Mathematics Subject Classification: 60H15
Key Words and Phrases: stochastic partial differential equations, nonlocal operators

⋅ 5th-D-09:30 − 10:00 [CANCELLED] Strong approximation of constrained stochastic dynamics (Felix Lindner, Holger Stroot)

Felix Lindner*, University of Kassel, Holger Stroot, Fraunhofer Institute for Industrial Mathematics ITWM

In this talk, a strong approximation result for a class of stochastic mechanical systems with nonlinear holonomic constraints is presented. Such systems are described by higher-index stochastic differential-algebraic equations, involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, whose coefficients are in general not globally monotone and of super-linear growth. Strong convergence is established for a half-explicit drift-truncated Euler scheme which fulfills the constraint exactly. Concrete examples for the considered systems are bead-rod chain models used in molecular dynamics as well as spatially discretized models for the dynamics of inextensible fibers in turbulent flows as occurring, e.g, in the spunbond production process of non-woven textiles.

2010 Mathematics Subject Classification: Primary 60H35, 74Hxx; Secondary 60H10, 58J65, 65C30
Key Words and Phrases: stochastic differential-algebraic equation, manifold-valued stochastic differential equation, nonlinear constraint, numerical approximation, drift-truncated scheme, strong convergence

⋅ 6th-G-13:30 − 15:30   Chair: Moritz Kassmann (Bielefeld University)

⋅ 6th-G-13:30 − 14:00 Aldous Markov chain on cladograms in the diffusion limit (Wolfgang Loehr, Leonid Mytnik, Anita Winter)

Wolfgang Loehr, University of Duisburg-Essen, Leonid Mytnik, Technion Haifa, Anita Winter*, University of Duisburg-Essen

In [1], Aldous investigates a symmetric Markov chain on cladograms and gives bounds on its mixing and relaxation times. The latter bound was sharpened in [2]. In this talk we encode cladograms as binary, algebraic measure trees and show that this Markov chain on cladograms with fixed number of leaves converges in distribution as the number of leaves goes to infinity. We give a rigorous construction of the limit, whose existence was conjectured by Aldous, as the solution of a well-posed martingale problem. We show that the Aldous diffusion is a Feller process with continuous paths, and the algebraic measure Brownian CRT is its unique invariant distribution. Furthermore, we consider the vector of the masses of the three subtrees connected to a sampled branch point. In the Brownian CRT, its annealed law is known to be the Dirichlet distribution. Here we give an explicit expression for the infinitesimal evolution of its quenched law under the Aldous diffusion. \begin{thebibliography}{9} \bibitem{1} David Aldous, {\it Mixing Time for a Markov Chain on Cladograms}, Combinatorics, Probability and Computing, 2000. \bibitem{2} Jason Schweinsberg, {\it An {$O(n\sp 2)$} bound for the relaxation time of a Markov chain on cladograms}, Random Structures Algorithms, 2001. \bibitem{3} Wolfgang Loehr, {\it Leonid Mytnik and Anita Winter}, arXiv:1805.12057. \end{thebibliography}

2010 Mathematics Subject Classification: Primary 60B99; Secondary 60G99, 60J05, 60J25, 60J60, 60J80
Key Words and Phrases: tree-valued Markov chain, tree-valued diffusion, algebraic trees, sample shape convergence, Gromov-weak convergence, Wright-Fisher diffusion, martingale problem, continuum tree

⋅ 6th-G-14:00 − 14:30 A well-posedness theory in Sobolev space for the stochastic Magnetohydrodynamic equations in the whole space (Il Doo Kim, Minsuk Yang)

김일두*((고려대)), 양민석((연세대))
Il Doo Kim*, Korea University, Minsuk Yang, Yonsei University

Taking the Leray projection $\mathbb{P}$ to the following stochastic MHD equation \begin{equation*} \left\{ \begin{aligned} &(\partial_t - \Delta) \mathbf{u} + \nabla\cdot(\mathbf{u}\otimes\mathbf{u}-\mathbf{b}\otimes\mathbf{b}) + \nabla \pi = g^k_1 \frac{dw^k}{dt} \\ &(\partial_t - \Delta) \mathbf{b} + \nabla\cdot(\mathbf{u}\otimes\mathbf{b}-\mathbf{b}\otimes\mathbf{u}) = g^k_2 \frac{dw^k}{dt} \\ &\nabla \cdot \mathbf{u} = \nabla \cdot \mathbf{b} = 0 \end{aligned} \right., \end{equation*} we formally have \begin{equation} \label{pro eqn} \left\{ \begin{aligned} &(\partial_t - \Delta) \mathbf{v} = - \mathbb{P}\nabla\cdot(\mathbf{w}\otimes\mathbf{v}) + G^k_1 \frac{dw^k}{dt} \\ &(\partial_t - \Delta) \mathbf{w} = - \mathbb{P}\nabla\cdot(\mathbf{v}\otimes\mathbf{w}) + G^k_2 \frac{dw^k}{dt} \\ &\nabla \cdot \mathbf{v} = \nabla \cdot \mathbf{w} = 0, \end{aligned} \right., \end{equation} where \begin{equation*} \mathbf{v} = \mathbf{u} + \mathbf{b}, \quad \mathbf{w} = \mathbf{u} - \mathbf{b}. \end{equation*} We will say that $(\mathbf{v}, \mathbf{w})$ is a (mild) solution to (1) on $(0,T)$ if it solves for $0\le t < T$ the integral equations \begin{equation*} \begin{split} \mathbf{v} & = S(t)\mathbf{v}_0-B(\mathbf{w},\mathbf{v}) +\int_0^t S(t-s) G^k_1(s)dw_s\\ \mathbf{w} & = S(t)\mathbf{w}_0 - B(\mathbf{v},\mathbf{w})+\int_0^t S(t-s) G^k_1(s)dw_s. \end{split} \end{equation*} In this talk, we find optimal regularity conditions on the initial values $(\mathbf{v}_0, \mathbf{w}_0)$ and stochastic external forces $(G_1,G_2)$ to guarantee the existence of a mild solution $(\mathbf{v}, \mathbf{w})$ with small initial data or within small time $T_0$. This is joint work with Minsuk Yang.

2010 Mathematics Subject Classification: 60H15, 35Q35
Key Words and Phrases: stochastic partial differential equations, magnetohydrodynamic equations

⋅ 6th-G-14:30 − 15:00 Metastability and effective interfaces for the high-intensity Widom-Rowlinson model (Frank den Hollander, Sabine Jansen, Roman Kotecky, Elena Pulvirenti)

Frank den Hollander, Leiden University, Sabine Jansen*, Ludwig Maximilian University of Munich, Roman Kotecky, University of Warwick, Elena Pulvirenti, University of Bonn

Consider a diffusion $X_t$ in an energy landscape $U(x)$, i.e., a solution to the stochastic differential equation $d X_t = - \nabla U(X_t) d t + \sqrt{\epsilon} d B_t$. In the small-noise limit $\epsilon \searrow 0$, the diffusion started in a local energy minimum exhibits metastable behavior - it takes a long time to reach the global minimum. The answer to the question ``how long" is provided by the Eyring-Kramers law. The talk addresses similar questions for a Markov birth and death process of points in $\mathbb R^d$, where the energy landscape is replaced with the rate function of some suitable large deviations principle and the analogue of the Eyring-Kramers law brings in functional central limit theorems and infinite-dimensional Gaussians. Based on joint work in progress with Frank den Hollander, Roman Kotecky and Elena Pulvirenti.

2010 Mathematics Subject Classification: 60J45, 60J60, 60K35
Key Words and Phrases: large deviations, Markov processes, metastability

⋅ 6th-G-15:00 − 15:30 Variational principles for discrete maps (Georg Menz)

Georg Menz, University of California-Los Angeles

About joint work with Martin Tassy and/or Andrew Krieger. Previous works have shown that arctic circle phenomenons and limiting behaviors of some integrable discrete systems can be explained by a variational principle. In this talk we present a method to deduce variational principles for non-integrable discrete systems. We illustrate the method by considering two different models. In the first model, we consider graph homomorphisms from $Z^d$ to a regular tree. In the second model, we derive a quenched variational principle for height functions exposed to a random field.

2010 Mathematics Subject Classification: 82B20, 82B30, 82B41
Key Words and Phrases: variational principles, non-integrable models, limit shapes, domino tilings, entropy, local surface tension, homogenization, sub-additive ergodic theorem

⋅ 6th-H-16:00 − 17:30   Chair: Panki Kim (Seoul National University)

⋅ 6th-H-16:00 − 16:30 Harnack inequality for balanced random walks (Jean Dominique Deuschel)

Jean Dominique Deuschel, Technical University of Berlin

We derive an ellipitic Harnack inequality for a balanced random walk in i.i.d. possibly degenerate random environment. We use both analytical and probabilistic tools. Joint work with N. Berger, M. Cohen and X. Guo.

2010 Mathematics Subject Classification: 60J27
Key Words and Phrases: random walks in random environment

⋅ 6th-H-16:30 − 17:00 On stable jump-diffusions (Leif D\"oring)

Leif D\"oring, University of Mannheim

We discuss recent computations for stable processes using the representation of stable processes as self-similar Markov processes. Transformations using Markov additive processes and the Wiener-Hopf transformation allow to deduce many explicit formulas. We present applications to the boundary behavior of stable processes.

2010 Mathematics Subject Classification: 60J75
Key Words and Phrases: stable processes, stable jump diffusions

⋅ 6th-H-17:00 − 17:30 Green kernel asymptotics for two-dimensional random walks under random conductances (Sebastian Andres, Jean-Dominique Deuschel, Martin Slowik)

Sebastian Andres*, University of Cambridge, Jean-Dominique Deuschel, Technical University of Berlin, Martin Slowik, Technical University of Berlin

The random conductance model is a well-established model for a random walk in random environment. In recent years the behaviour of the associated heat kernel and Green function has been intensively studied, and in dimension $d\geq 3$ the asymptotics of the Green kernel are meanwhile quite well-understood. In this talk we present precise asymptotics of the potential kernel and the Green function of the walk killed upon exiting balls in dimension $d=2$. This result holds, for instance, in the case of strictly elliptic conductances, random walks on supercritical percolation clusters or ergodic degenerate conductances satisfying a moment condition.

2010 Mathematics Subject Classification: 39A12, 60J35, 60K37, 82C41
Key Words and Phrases: random conductance model, Green kernel
Inverse problems and imaging science

⋅ 4th-A-09:00 − 10:00   Chair: Jin Keun Seo (Yonsei University)

⋅ 4th-A-09:00 − 09:30 Monotonicity methods for inverse coefficient problems (Bastian Harrach)

Bastian Harrach, Goethe University Frankfurt

Newly emerging imaging methods lead to the inverse problem of determining one or several coefficient function(s) in an elliptic partial differential equation from (partial) knowledge of its solutions. A natural and generic approach is to approximate the unknown coefficients by minimizing a linearized and regularized data-fit functional. In this talk, we will show how to regularize the linearized data-fit functional with monotonicity-based constraints in such a way that convergence of certain shape properties in the reconstructions can be rigorously guaranteed.

2010 Mathematics Subject Classification: 35R30
Key Words and Phrases: inverse coefficient problems, electrical impedance tomography, monotonicity method

⋅ 4th-A-09:30 − 10:00 Application of MUSIC algorithm in microwave imaging (Won-Kwang Park)

박원광((국민대))
Won-Kwang Park, Kookmin University

In this contribution, we consider Multiple Signal Classification (MUSIC)-type algorithm for a non-iterative microwave imaging of small anomalies located in a homogeneous media from scattering matrix without diagonal elements. In order to explain the feasibility of MUSIC in microwave imaging, we investigate mathematical structure of MUSIC by establishing a relationship with an infinite series of Bessel function of integer order and antennas setting. This is based on the representation formula of scattering parameters in the presence of small anomalies and the application of Born approximation. Simulation results using real-data at $f=~$925MHz frequency are exhibited to show the feasibility of designed algorithm and to support investigated structure of imaging function.

2010 Mathematics Subject Classification: 65N21, 78A46
Key Words and Phrases: microwave imaging, Multiple Signal Classification (MUSIC), Bessel function, real-data experiments

⋅ 4th-B-13:30 − 14:30   Chair: Jin Keun Seo (Yonsei University)

⋅ 4th-B-13:30 − 14:00 Reduced basis methods for MREIT (Dominik Garmatter, Bastian Harrach)

Dominik Garmatter*, Chemnitz University of Technology, Bastian Harrach, Goethe University Frankfurt

The numerical solution of parameter identification problems in a partial differential equation (PDE) setting from (noisy) measurements usually requires numerous amounts of forward solutions of the respective PDE. One way to speed up the solution process therefore is to reduce the computational time of the forward solution, e.g. via the reduced basis method. The reduced basis method is a model order reduction technique which constructs a low-dimensional subspace of the solution space. Galerkin projection onto that space allows for an approximative solution. An efficient offline/online decomposition enables the rapid computation of the approximative solution for many different parameters. This talk will focus on the problem of magnet resonance electrical impedance tomography (MREIT), where the main objective is the acceleration of the well-known Harmonic $B_z$ Algorithm [1] using the adaptive reduced basis framework developed in [2]. The general idea of the framework is to adaptively construct a small, problem-oriented reduced basis space instead of constructing a global reduced basis space like it is usually the case in reduced basis methods. This will be done in an iterative procedure: the Harmonic $B_z$ Algorithm is projected onto the current reduced basis space and iterated until certain termination criteria are reached. The resulting parameter then is utilized to enrich the reduced basis space and therefore fit it to the given problem. This process is repeated until an iterate is accepted as the solution of the inverse problem. Numerical results will demonstrate the usefulness of the approach. \begin{thebibliography}{9} \bibitem{1} D. Garmatter, B. Haasdonk, and B. Harrach, {\it A Reduced Basis Landweber method for nonlinear inverse problems}, Inverse Problems {\bf 32} (2016), no. 3, 035001. \bibitem{2} J. K. Seo, J.-R. Yoon, E. J. Woo, and O. Kwon, {\it Reconstruction of conductivity and current density images using only one component of magnetic field measurements}, IEEE Transactions on Biomedical Engineering {\bf 50} (2003), no. 9, 1121--1124. \end{thebibliography}

2010 Mathematics Subject Classification: 35R30, 35R05, 65N21
Key Words and Phrases: MREIT, image reconstruction, convergence, reduced basis method, model order reduction, adaptive space generation

⋅ 4th-B-14:00 − 14:30 Deep learning based method for metal artifact correction in X-ray CT (Hyoung Suk Park)

박형석((국가수리과학연구소))
Hyoung Suk Park, NIMS

Recently, deep learning techniques show significant performance over existing methods for medical imaging modalities including X-ray computed tomography (CT). In this talk, I will introduce how deep learning is utilized to reduce metal artifacts in X-ray CT imaging. Taking account of the fundamental difficulty in obtaining sufficient training data in a medical environment, the proposed learning method is designed to use simulated training data and a patient-type specific learning model is used to simplify the learning process. I will show the feasibility of the proposed method using numerical simulations and phantom experiment.

2010 Mathematics Subject Classification: 68W01
Key Words and Phrases: computed tomography, metal artifact reduction, deep learning

⋅ 4th-B-14:30 − 15:30   Chair: Kiwan Jeon (NIMS)

⋅ 4th-B-14:30 − 15:00 Convex regularization of discrete-valued inverse problems (Christian Clason, Thi Bich Tram Do, Florian Kruse, Karl Kunisch)

Christian Clason*, University Duisburg-Essen, Thi Bich Tram Do, University Duisburg-Essen, Florian Kruse, University of Graz, Karl Kunisch, University of Graz

This talk is concerned with parameter identification problems where a distributed parameter is known a priori to only take on values from a given discrete set. This property can be promoted in Tikhonov regularization with the aid of a suitable convex but nondifferentiable regularization term. This allows applying standard approaches to show well-posedness and convergence rates in Bregman distance. Using the specific properties of the regularization term, it can be shown that convergence (albeit without rates) actually holds pointwise. Furthermore, the resulting Tikhonov functional can be minimized efficiently using a semi-smooth Newton method. Numerical examples illustrate the properties of the regularization term and the numerical solution. We also address the combination with total variation regularization.

2010 Mathematics Subject Classification: 49N45, 65N21
Key Words and Phrases: parameter identification, regularization, non-smooth optimization

⋅ 4th-B-15:00 − 15:30 Image colorization with nonlocal and local regularizers (Myeongmin Kang, Myungjoo Kang, Miyoun Jung)

강명민((충남대)), 강명주((서울대)), 정미연*((한국외국어대))
Myeongmin Kang, Choongnam National University, Myungjoo Kang, Seoul National University, Miyoun Jung*, Hankuk University of Foreign Studies

In this talk, we introduce a novel model for the restoration of a color image from a grayscale image with color data given only in small regions. The model is based on the idea of the generalization of the low dimensional manifold model and the YCbCr color space. It involves two prior terms, a weighted nonlocal Laplacian (WNLL) and a weighted total variation (WTV). The WNLL allows regions without color information to be interpolated smoothly from given sparse color data, while the WTV assists to inhibit the diffusion of color values across edges. To handle various types of sampled data, we introduce an updating rule for the weight function in the WNLL. Besides, we present an efficient iterative algorithm for solving the proposed model, and we show numerical results to validate the superior performance of the proposed model over that of the other state-of-the-art models.

2010 Mathematics Subject Classification: 68U10
Key Words and Phrases: image colorization, nonlocal method, low dimensional manifold, weighted total variation

⋅ 5th-E-13:30 − 15:30   Chair: Bastian Harrach (Goethe University Frankfurt)

⋅ 5th-E-13:30 − 14:00 Some applications using a deep learning (Myungjoo Kang, Jihoon Kwak, Hyunkook Kim)

강명주*((서울대)), 곽지훈((서울대)), 김현욱((서울대))
Myungjoo Kang*, Seoul National University, Jihoon Kwak, Seoul National University, Hyunkook Kim, Seoul National University

In this presentations, I will show some interesting developments using the deep learning techniques. Those cover music generations, beauty scores and news recommendations.

2010 Mathematics Subject Classification: 68T01
Key Words and Phrases: deep learning, image analysis

⋅ 5th-E-14:00 − 14:30 Feature extraction and artefact reduction in dynamic imaging (Melina Kienle Garrido, Bernadette Hahn)

Melina Kienle Garrido*, University of Wuerzburg, Bernadette Hahn, University of Wuerzburg

Imaging modalities represent a well-known application of the theory of inverse problems. In this context, it is typically assumed that the sought-for functions are independent of time. However, this assumption is often violated, e.g. in medical applications due to patient movements. In this case, standard reconstruction techniques lead to motion artefacts in the computed images. To compensate for the motion implies to include the time-dependency by suitable modelling and specially designed reconstruction algorithms. For example dynamic photoacoustic tomography and dynamic computerized tomography can be modeled by incorporating a displacement model in the static forward operator. If the Fourier integral structure of the operator is preserved, methods from microlocal analysis can provide insights on the design of the reconstruction algorithms. In this talk, we present an effective method to extract features of an object from motion-corrupted data with significantly reduced motion artefacts.

2010 Mathematics Subject Classification: 92C55, 35S30, 58J40
Key Words and Phrases: dynamic imaging, inverse problems, feature extraction

⋅ 5th-E-14:30 − 15:00 A learning-based method for solving ill-posed nonlinear inverse problems (Jin Keun Seo)

서진근((연세대))
Jin Keun Seo, Yonsei University

This talk presents new approach for solving ill-posed nonlinear inverse problems. For ease of explanation of the proposed approach, we use the example of lung electrical impedance tomography (EIT), which is known to be a nonlinear and ill-posed inverse problem. Conventionally, penalty-based regularization methods have been used to deal with the ill-posed problem. However, experiences over the last three decades have shown methodological limitations in utilizing prior knowledge about tracking expected imaging features for medial diagnosis. The proposed method's paradigm is a completely different from the conventional approaches; the proposed reconstruction uses a variety of training data sets to generate a low dimensional manifold of approximate solutions, which allows to convert the ill-posed problem to a well-posed one. Variational autoencoder was used to produce a compact and dense representation for lung EIT images with a low dimensional latent space. Then, we learn a robust connection between the EIT data and the low-dimensional latent data. Numerical simulations validate the effectiveness and feasibility of the proposed approach.

2010 Mathematics Subject Classification: 45Q05
Key Words and Phrases: deep learning, inverse problem, EIT

Algebra

⋅ 5th-D-09:00 − 10:00   Chair: Bo-Hae Im (KAIST)

⋅ 5th-D-09:00 − 09:30 Finiteness of real quadratic number fields which admit positive definite septenary universal lattices (Byeong Moon Kim, Myung-Hwan Kim, Dayoon Park)

김병문((강릉원주대)), 김명환((서울대)), 박다윤*((서울대))
Byeong Moon Kim, Gangneung-Wonju National University, Myung-Hwan Kim, Seoul National University, Dayoon Park*, Seoul National University

Universality is one of the most important topics in the theory of quadratic forms. For example, the famous Lagrange's four square theorem says that every positive rational integer is represented by the sum of four squares. In 1996, all positive definite ternary universal quadratic forms over real quadratic number fields were found by W. K. Chan, M.-H. Kim and S. Raghavan. In 2000, B. M. Kim proved that there are infinitely many real quadratic number fields which admit positive definite octonary universal quadratic forms. But it is unknown whether 8 is the minimal rank with this property. In this talk, we will prove that there exist only finitely many real quadratic number fields which admit a positive definite septenary universal quadratic forms. This implies that 8 is indeed the minimal rank with the property. This is a joint work with B. M. Kim and M.-H. Kim.

2010 Mathematics Subject Classification: 11E12
Key Words and Phrases: universal septenary quadratic lattice over quadratic number fields

⋅ 5th-D-09:30 − 10:00 A parametrization of $\theta$-congruent numbers with many prime factors (Hansol Kim, Bo-Hae Im)

김한솔*((한국과학기술원)), 임보해((한국과학기술원))
Hansol Kim*, KAIST, Bo-Hae Im, KAIST

In this talk, we give a parametrization of $\theta$-congruent numbers with many prime factors including any given primes by showing the positivity of the rank of the corresponding $\theta$-congruent number elliptic curve over $\mathbb{Q}$. In particular, we compute the probability of the set of primes dividing the parametrization, which shows that a given prime $p>2n$ close to $2n$ where $n$ is related with the degree of the parametrization appears as a factor very often. This is a joint work with Bo-Hae Im.

2010 Mathematics Subject Classification: 11G05
Key Words and Phrases: congruent number, elliptic curve, $\theta$-congruent number

⋅ 6th-G-13:30 − 14:30   Chair: Donghi Lee (Pusan National University)

⋅ 6th-G-13:30 − 14:00 [Invited Talk of MSJ] Enriques surfaces with finite automorphism group (Shigeyuki Kondo)

Shigeyuki Kondo, Nagoya University

An Enriques surface is a non rational surface which occupies a block of the classification table of algebraic surfaces. F. Enriques (1907) showed that a generic Enriques surface has an infinite, non-cyclic, discontinuous group of automorphisms, and presented a question about the existence of an Enriques surface with only finitely many automorphisms. In 1911, G. Fano gave an example of Enriques surfaces with finite automorphism group. Later, by using the Torelli type theorem for K3 surfaces, V. Nikulin and S. Kondo classified all complex Enriques surfaces　with finite automorphism group (1984, 1986). Here we remark that any complex Enriques surface is the quotient of a K3 surface by a fixed point free involution. Recently the classification of all Enriques surfaces with finite automorphism group in any characteristics has been obtained by T. Katsura, S. Kondo and G. Martin. In this talk I will discuss the problem and a key idea of our approach.

2010 Mathematics Subject Classification: 14J28
Key Words and Phrases: Enriques surfaces, automorphism group

⋅ 6th-G-14:00 − 14:30 Orthogonality for exact and approximate Latin squares (Bokhee Im, Jonathan D. H. Smith)

임복희*((전남대)), Jonathan D. H. Smith((Iowa State University))
Bokhee Im*, Chonnam National University, Jonathan D. H. Smith, Iowa State University

Questions concerning mutually orthogonal Latin squares and finite projective planes have previously been addressed by various combinatorial methods. We now propose a study of these questions by geometric methods, based on an analysis of the polytope of tristochastic tensors, and the real projective space of lines around the barycenter of the polytope. Using homogeneous coordinates in this projective geometry, orthogonality of Latin squares and quasigroups reduces to the usual concept of orthogonality of vectors. Points in the projective geometry corresponding to exact Latin squares are identified in the geometry by a partition condition.

2010 Mathematics Subject Classification: 05B15, 20N05, 51F20
Key Words and Phrases: Birkhoff polytope, stochastic matrix, quasigroup, Latin square, approximate Latin square, tristochastic tensor

⋅ 6th-G-14:30 − 15:30   Chair: Youn-Seo Choi (KIAS)

⋅ 6th-G-14:30 − 15:00 Direct sums of endoprime modules (Gangyong Lee, S. Tariq Rizvi)

이강용*((충남대)), S. Tariq Rizvi((The Ohio State University at Lima))
Gangyong Lee*, Chungnam National University, S. Tariq Rizvi, The Ohio State University at Lima

The study of prime rings and prime ideals has been an important topic in Ring Theory because these notions help provide the description of structures of rings and modules related to them. A ring $R$ is said to be \emph{prime} if every nonzero left ideal is faithful as a left $R$-module. In 2005, Haghany and Vedadi defined the notion of an endoprime module which generalizes the notion of a prime ring to a general module theoretic setting. Recall that a module $M$ is said to be \emph{endoprime} if any nonzero fully invariant submodule of $M$ is faithful as a left module over $S=\text{End}_R(M)$. Equivalently, $\mathbf{l}_S(N)=0$ for all $0 \neq N \unlhd M$. In this talk, we further study endoprime modules and present a complete characterization of when an arbitrary direct sum of endoprime modules is also endoprime although even finite direct sums of endoprime modules do not inherit the property. Also, we fully characterize an endoprime module in terms of its endomorphism ring. The endoprime property is useful in view of its role in the Morita contexts which arise from an endoprime module. Applications of our results and examples illustrating and delimiting the results are provided.

2010 Mathematics Subject Classification: 16D70, 16S50, 16N60
Key Words and Phrases: endoprime module, FI-indecomposable module, semicentral reduced ring, column (and row) finite matrix ring

⋅ 6th-G-15:00 − 15:30 On skew power-serieswise Armendariz rings and skew IFP rings (Nam Kyun Kim)

김남균((한밭대))
Nam Kyun Kim, Hanbat National University

In this paper we study the structures of power-serieswise Ar- mendariz rings and IFP rings when they are skewed by ring endomorphisms (or automorphisms). We call such rings skew power-serieswise Armendariz rings and skew IFP rings, respectively. We also investigate relationships among them and construct necessary examples in the pro- cess. The results argued in this note can be extended to the ordinary ring theoretic properties of power-serieswise Armendariz rings, IFP rings, and near-related rings.

2010 Mathematics Subject Classification: 16S36
Key Words and Phrases: zer-divisor, skew power-serieswise Armendariz ring, skew IFP ring

⋅ 6th-H-16:00 − 17:00   Chair: Jung Wook Lim (Kyungpook National University)

⋅ 6th-H-16:00 − 16:30 McCoy modules and related modules over commutative rings (Dan Anderson, Sangmin Chun)

Dan Anderson((University of Iowa)), 천상민*((중앙대))
Dan Anderson, University of Iowa, Sangmin Chun*, Chung Ang University

Let $M$ be a left $R$-module. Then $M$ is a McCoy (resp., dual McCoy) module if for nonzero $f (X) \in R[X]$ and $m(X)\in M[X]$, $f (X)m(X) = 0$ implies there exists a nonzero $r\in R$ (resp., $m \in M$) with $rm(X) = 0$ (resp., $f (X)m = 0$). We show that for $R$ commutative every $R$-module is dual McCoy, but give an example of a non-McCoy module. A number of other results concerning (dual) McCoy modules as well as arithmetical, Gaussian, and Armendariz modules are given.

2010 Mathematics Subject Classification: Primary 13C13; Secondary 16D80
Key Words and Phrases: arithmetical module, Armendariz module, dual McCoy module, Gaussian module, McCoy module

⋅ 6th-H-16:30 − 17:00 An explicit matrix factorization of cubics of small dimension (Yeongrak Kim, Frank-Olaf Schreyer)

김영락*((University of Saarland)), Frank-Olaf Schreyer((University of Saarland))
Yeongrak Kim*, University of Saarland, Frank-Olaf Schreyer, University of Saarland

An early concept of matrix factorizations can be found in physics, especially by Dirac on the study of Heisenberg matrix mechanics. Eisenbud introduced this notion in mathematics to study homological algebra over hypersurface rings. Recently, matrix factorizations are getting more significant due to their various applications in commutative algebra, algebraic geometry, and mathematical physics. When a matrix factorization is induced by a linear matrix, it gives a presentation of an Ulrich sheaf which appears naturally in a number of important questions. Nevertheless, very few is known about Ulrich sheaves. In particular, the smallest possible rank of an Ulrich sheaf on a general cubic fourfold is unknown. Manivel recently shows that there are rank 9 Ulrich sheaves on a general cubic sevenfold using the invariant theory for the Lie group $E_6$. In this talk, I will give an alternative proof using Shamash's construction with the spinor variety. In particular, this provides the explicit description of a certain matrix factorization of the Cartan cubic. I will also give the complete classification of cubic forms whose Hessian induce matrix factorization of themselves via XJC-correspondence of Pirio and Russo.

2010 Mathematics Subject Classification: 13D02, 13P05, 14Q10
Key Words and Phrases: matrix factorization, Ulrich sheaf, Shamash's construction, Spinor variety, Cartan cubic, XJC-correspondence

⋅ 6th-H-17:00 − 17:50   Chair: Jaebum Sohn (Yonsei University)

⋅ 6th-H-17:00 − 17:20 Note on t-Schreier monoid domains (Kiwoong Kwon, Jungwook Lim)

권기웅*((경북대)), 임정욱((경북대))
Kiwoong Kwon*, Kyungpook National University, Jungwook Lim, Kyungpook National University

Let $R$ be an integral domain and $M$ a nontrivial cancellative torsion-free commutative monoid. If the group of $t$-invertible $t$-ideals of satisfies the Riesz interpolation property, $R$ is called $t$-Schreier. Similarly, $M$ is called $t$-Schreier provided that the group of $t$-invertible $t$-ideals of satisfies the Riesz interpolation property. We show that the following properties of a commutative monoid domain $R[M]$ are equivalent: (1) $R[M]$ is $t$-Schreier; (2) $R$ and $M$ are t-Schreier,

2010 Mathematics Subject Classification: 13A15
Key Words and Phrases: star operation, Schreier domain

⋅ 6th-H-17:30 − 17:50 Basic properties of $w$-SFT-rings (Minjae Kwon, Jungwook Lim)

권민재*((경북대)), 임정욱((경북대))
Minjae Kwon*, Kyungpook National university, Jungwook Lim, Kyungpook National university

Let $R$ be an integral domain with quotient field $K$. We define a nonzero ideal $A$ of $R$ to be a $w$-SFT-ideal if there exist a finitely generated ideal $B \subseteq A$ and a positive integer $k$ such that $a^k \in B_w$ for each $a \in A_w$. The ring $R$ is said to be a $w$-SFT-ring if each nonzero ideal of $R$ is a $w$-SFT-ideal. Note that an ideal $A$ of $R$ is a $w$-SFT ideal if and only if $A_w$ is a $w$-SFT-ideal. In this talk, we study the basic properties of $w$-SFT-rings. We show that an integral domain $R$ is a Krull domain if and only if $R$ is a completely integrally closed $w$-SFT-ring.

2010 Mathematics Subject Classification: 13A15
Key Words and Phrases: Krull domain, Star operation
Analysis I

⋅ 4th-C-16:00 − 18:00   Chair: Jaeseong Heo (Hanyang University)

⋅ 4th-C-16:00 − 16:30 Weighted Fourier algebra, Gelfand spectrum and complexification of Lie groups (Hun Hee Lee)

이훈희((서울대))
Hun Hee Lee, Seoul National University

We investigate Beurling-Fourier algebras, a weighted version of Fourier algebras, on various Lie groups focusing on their spectral analysis. We will introduce a refined general definition of weights on the dual of locally compact groups and their associated Beurling-Fourier algebras. Constructions of nontrivial weights will be presented focusing on the cases of representative examples of Lie groups, namely $SU(n)$, the Heisenberg group $\mathbb{H}$, the reduced Heisenberg group $\mathbb{H}_r$, the Euclidean motion group $E(2)$ and its simply connected cover $\widetilde{E}(2)$. We will determine the spectrum of Beurling-Fourier algebras on each of the aforementioned groups emphasizing its connection to the complexification of underlying Lie groups. We also demonstrate ``polynomially growing" weights does not change the spectrum and show the associated regularity of the resulting Beurling-Fourier algebras.

2010 Mathematics Subject Classification: 47L25, 46B07
Key Words and Phrases: Fourier algebra, operator algebra, Beurling algebra, Gelfand spectrum, complexification of Lie groups

⋅ 4th-C-16:30 − 17:00 $\mathcal{J}$-Weyl's theorem for Krein space self-adjoint operators (Il Ju An, Jaeseong Heo)

안일주*((이화여대)), 허재성((한양대))
Il Ju An*, Ewha Womans University, Jaeseong Heo, Hanyang University

The purpose of this talk is to introduce Krein space self-adjoint operators and new classes of $\mathcal{J}$-Fredholm, $\mathcal{J}$-Weyl, and $\mathcal{J}$-Browder operators. Also, we define some spectra of such operators, $\mathcal{J}$-Weyl's theorem, and $\mathcal{J}$-Browder's theorem. Moreover, we explain the relations between such theorems and investigate that $\mathcal{J}$-Weyl's theorem or $\mathcal{J}$-Browder's theorem, respectively, holds for Krein space self-adjoint operators if and only if it holds for their dual operators.

2010 Mathematics Subject Classification: 47A10, 47A11, 47A53, 47B50
Key Words and Phrases: Krein space, J-kernel, J-Fredholm index, essential spectrum, J-Weyl spectrum

⋅ 4th-C-17:00 − 17:30 On numerical ranges of conjugations and antilinear operators (Injo Hur, Ji Eun Lee)

허인조((울산과학기술원)), 이지은*((세종대))
Injo Hur, UNIST, Ji Eun Lee*, Sejong University

In this paper, we investigate the numerical ranges of conjugations and antilinear operators, which will be shown to be annuli in general. This result demonstrates that Toeplitz-Hausdorff Theorem, which says the convexity on the numerical ranges of linear operators, does not hold for the ones of antilinear operators.

2010 Mathematics Subject Classification: Primary 47A05; Secondary 47A12
Key Words and Phrases: antilinear operator, conjugation, numerical range

⋅ 4th-C-17:30 − 18:00 Entropic uncertainty relations under localizations on discrete quantum groups (SangGyun Youn)

윤상균((서울대))
SangGyun Youn, Seoul National University

The uncertainty principle implies that both a non-zero function $f$ and its Fourier transform $\widehat{f}$ cannot be sharply localized. In other words, if a function $f$ is concentrated, then its Fourier transform $\widehat{f}$ should be dispersed. This principle has been explored within the framework of locally compact quantum groups in recent years. In this talk, we demonstrate that entropic uncertainty relations can be strengthened under localizations on discrete quantum groups.

2010 Mathematics Subject Classification: 46L89, 81R15
Key Words and Phrases: entropic uncertainty principle, localization, free orthogonal quantum group and lacunary sets

⋅ 5th-D-09:00 − 10:00   Chair: Hun Hee Lee (Seoul National University)

⋅ 5th-D-09:00 − 09:30 Wasserstein metric and Wasserstein mean of positive definite matrices (Jinmi Hwang, Sejong Kim)

황진미*((충북대)), 김세정((충북대))
Jinmi Hwang*, Chungbuk National University, Sejong Kim, Chungbuk National University

Recently the new Riemannian metric on the cone of positive definite Hermitian matrices, called the Wasserstein metric, has been introduced. We see its background, meanings in various areas, and several properties. Furthermore, we investigate the least squares mean with respect to the Wasserstein metric, called the Wasserstein mean, and explore some properties such as the determinantal inequality and bounds with respect to the operator norm and Loewner order. Finally we suggest its generalization to two points of view, the parameterized metric with sandwiched quasi-relative entropy and positive invertible operators.

2010 Mathematics Subject Classification: 15B48
Key Words and Phrases: Wasserstein metric, Wasserstein mean

⋅ 5th-D-09:30 − 10:00 A necessary and sufficient condition for the existence of invariant Gibbs measures (Minkyu Kim, Uijin Jung)

김민규*((아주대)), 정의진((아주대))
Minkyu Kim*, Ajou University, Uijin Jung, Ajou University

An invariant Gibbs measure is a special equilibrium state and is used in many fields of mathematics including thermodynamic formalism of topological dynamics. \par It is well known that if $X$ is a mixing shift of finite type and $f$ is a H\"{o}lder continuous function from $X$ to $\mathbb{R}$, then there exists a unique invariant Gibbs measure with respect to the potential $f$ which is Bernoulli. Bowen proved that if $X$ is a subshift with the specification property and $f$ is a function in the Bowen class, then there exists a unique equilibrium state for $f$ which is a Gibbs measure for $f$. These results do not give much information outside specification. Baker and Ghenciu studied the existence of Gibbs measures for the potential $0$ in general shift spaces. They showed that there exists a Gibbs measure for the potential $0$ if and only if $X$ is (right-)balanced. Since the constructed measure need not be invariant in general and one working on dynamics and ergodic theory is usually interested in invariant measures, we study a necessary and sufficient condition for the existence of invariant Gibbs measures for the potential $0$ and generalize this result for real-valued continuous functions of $X$. \par Let $X$ be a subshift and let $f$ be a real-valued continuous function of $X$. We say that $X$ is right-balanced with respect to $f$ if there exists $c>0$ such that for $\forall u\in\mathcal{B}(X), \forall x_{uv}\in[uv], \forall x_u\in[u],$ and $\forall x_w\in[w]$, $$c^{-1}\le\frac{\sum_{\substack{v\in\mathcal{B}_n(X)\\uv\in\mathcal{B}(X)}}{\exp(S_{\vert u\vert+n}f(x_{uv}))}}{\exp(S_{\vert u\vert}f(x_u))\sum_{w\in\mathcal{B}_n(X)}{\exp(S_nf(x_w))}}\le c,$$ where $m=\vert u\vert$. Similarly, we can define left-balancedness w.r.t. $f$. Using balancedness, we show that a subshift $X$ has an invariant Gibbs measure for the potential $f$ if and only if $X$ is right and left-balanced w.r.t. $f$ and that a subshift $X$ with the specification property has an invariant Gibbs measure for the potential $f$ if and only if $f$ is in the Bowen class.

2010 Mathematics Subject Classification: 37B10,37D35
Key Words and Phrases: subshifts, Gibbs measures, equilibrium states, thermodynamic formalism

⋅ 6th-F-09:00 − 10:00   Chair: Doyoon Kim (Korea University)

⋅ 6th-F-09:00 − 09:30 Existence and asymptotics of solutions of the self-dual O(3) Maxwell-Chern-Simons-Higgs equations (Jongmin Han)

한종민((경희대))
Jongmin Han, Kyung Hee University

In this talk, we give recent results on the self-dual O(3) Maxwell-Chern-Simons-Higgs equations arising from planar condensed matter physics. The equations are divided into two categories: model for asymmetric vacua and model for symmetric vacua. From the physical motivation, we consider the equations on the whole two dimensional plane or the flat torus. We review recent progress on the existence and asymptotics of solutions for both cases, and give the main results on the existence of solutions on a tours for the symmetric vacua case.

2010 Mathematics Subject Classification: 35J61, 35Q75, 81T13
Key Words and Phrases: O(3) Maxwell-Chern-Simons-Higgs equations, condensate solutions

⋅ 6th-F-09:30 − 10:00 Variational construction of spike layer solutions for a singularly perturbed Neumann problem (Sang-hyuck Moon, Jaeyoung Byeon)

문상혁*((한국과학기술원)), 변재형((한국과학기술원))
Sang-hyuck Moon*, KAIST, Jaeyoung Byeon, KAIST

We consider the following singularly perturbed problem $\epsilon^2 \Delta u - u +f(u)=0, u>0$ in $\Omega$, $\frac{\partial u}{\partial \nu}=0$ on $\partial \Omega$ An existence of solutions with a spike layer near critical points of the mean curvature on the boundary $\partial \Omega$ is well known when a nondegeneracy for a limiting problem holds. We construct such solutions by a variational method which does not depend on the nondengeneracy for the limiting problem. By the variational approach, we construct the solutions for an optimal class of nonlinearities f satisfying the Berestycki-Lions conditions.

2010 Mathematics Subject Classification: 35J20, 35J60, 35B25, 35B40
Key Words and Phrases: singular perturbation, Neumann condition, spike layer, mean curvature, transplantation flow, gradient flow, variational

⋅ 6th-G-13:30 − 15:30   Chair: Jongmin Han (Kyung Hee University)

⋅ 6th-G-13:30 − 14:00 On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights (Hongjie Dong, Doyoon Kim)

Hongjie Dong((Brown University)), 김도윤*((고려대))
Hongjie Dong, Brown University, Doyoon Kim*, Korea University

We present mean oscillation estimates combined with Muckenhoupt weights. We then apply these results with Fefferman-Stein theorems on sharp functions to establish mixed-norm weighted $L_p$-estimates for divergence and non-divergence type elliptic and parabolic equations / systems with (partially) BMO coefficients in regular or irregular domains.

2010 Mathematics Subject Classification: 35R05, 42B37, 35B45, 35K25, 35J48
Key Words and Phrases: sharp/maximal functions, elliptic and parabolic equations, weighted Sobolev spaces, measurable coefficients

⋅ 6th-G-14:00 − 14:30 Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves (Blake Barker, Soyeun Jung, Kevin Zumbrun)

Blake Barker((Brigham Young University)), 정소연*((공주대)), Kevin Zumbrun((Indiana University))
Blake Barker, Brigham Young University, Soyeun Jung*, Kongju National University, Kevin Zumbrun, Indiana University

Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for viscous systems of conservation laws. Here, we derive conditions for Turing instability in viscous systems of conservation laws and use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime.

2010 Mathematics Subject Classification: 35L65
Key Words and Phrases: turing instability, viscous conservation laws, stable periodic waves

⋅ 6th-G-14:30 − 15:00 Asymptotic behavior and stability problem for the Schr\"{o}dinger-Lohe model (Hyungjin Huh, Seung-Yeal Ha, Dohyun Kim)

허형진((중앙대)), 하승열((서울대)), 김도현*((서울대))
Hyungjin Huh, Chung-Ang University, Seung-Yeal Ha, Seoul National University, Dohyun Kim*, Seoul National University

We present asymptotic behavior and stability problem for the Schr\"{o}dinger-Lohe(S-L) system which was first introduced as a possible phenomenological model exhibiting quantum synchronization. We present several sufficient frameworks leading to the emergent behaviors of the S-L system. To be more precise, we basically consider three types of network structures which describe the way how oscillators communicate with each others. In the presence of the network structures, we can expect interesting dynamical patterns other than synchronization, such as closed orbits and bi-polar states. In particular, under the all-to-all network, we show that there are only two possible asymptotic states: completely synchronized state or bi-polar state. Moreover, we claim that complete synchronization can happen for generic initial data showing that bi-polar state is unstable.

2010 Mathematics Subject Classification: 82C10, 82C22, 35B35
Key Words and Phrases: correlation function, emergence, stability, Kuramoto model, Schr\"{o}dinger–Lohe system, quantum synchronization

⋅ 6th-G-15:00 − 15:30 Dirichlet problems for elliptic equations with singular drifts in Lipschitz domains (Hyunseok Kim, Hyunwoo Kwon)

김현석((서강대)), 권현우*((서강대))
Hyunseok Kim, Sogang University, Hyunwoo Kwon*, Sogang University

We consider the Dirichlet problems for second-order linear elliptic equations $$-\triangle u+\mathrm{div}\left(u\mathbf{b}\right)=f,\quad-\triangle v-\mathbf{b}\cdot\nabla v=g $$ in a bounded Lipschitz domain $\Omega\subset\mathbb{R}^{n}$, $n\ge3$, where $\mathbf{b}:\Omega\rightarrow\mathbb{R}^{n}$ is given. Under the assumption $\mathbf{b}\in L^{n}(\Omega)^{n}$, we first establish the existence and uniqueness of solutions in $L_{\alpha}^{p}(\Omega)$. Here $L_{\alpha}^{p}(\Omega)$ denotes the Sobolev spaces with the pair $(\alpha,p)$ satisfying certain conditions. Based on this result, we study the unique solvability of Dirichlet problem when the boundary data is in $L^2(\partial\Omega)$.

2010 Mathematics Subject Classification: 35J15, 35J25
Key Words and Phrases: existence and uniqueness, elliptic equations, Lipschitz domains, singular drift terms

⋅ 6th-H-16:00 − 18:00   Chair: Jaeyoung Byeon (KAIST)

⋅ 6th-H-16:00 − 16:30 The property of the Lebesque space for special functions (Jaiok Roh)

노재옥((한림대))
Jaiok Roh, Hallym University

In this talk, we will consider Lebesque space for unbounded domain. And we will discuss the property which usually is valid for bounded domain. In this talk we will give the proof by detail and discuss some applications.

2010 Mathematics Subject Classification: 35J60
Key Words and Phrases: Lebesque space, incompressible flows, Euler equations

⋅ 6th-H-16:30 − 17:00 On isospectral canonical system flows (Injo Hur, Darren C. Ong)

허인조*((울산과학기술원)), Darren C. Ong((Xiamen University Malaysia))
Injo Hur*, UNIST, Darren C. Ong, Xiamen University Malaysia

We will revisit the structure of Korteweg-de Vries (KdV) flows, especially KdV hierarchy, whose spectra are invariant, when the time passes. This isospectral flows are related to the existence of solitary waves like Tsunami. Except a traveling wave which is the simplest example, KdV equation is the most well studied one having such solutions. Since KdV equations are naturally generalized to canonical systems, we may expect that iso-spectral canonical systems would be found. In this talk, we will, however, show that it will be very difficult for such canonical flows like KdV ones to exist. To see this, due to the lack of existence on operators related to canonical systems, so called zero curvature equation will be applied, which is equivalent to the famous Lax pair formalism.

2010 Mathematics Subject Classification: 34L40
Key Words and Phrases: canonical system, KdV hierarchy, isospectral

⋅ 6th-H-17:00 − 17:30 Remarks on some continuum models with diffusion derived from the synchronous particle models (Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee, Se Eun Noh)

하승열((서울대)), 김도현((서울대)), 이재승*((서울대)), 노세은((명지대))
Seung-Yeal Ha, Seoul National University, Dohyun Kim, Seoul National University, Jaeseung Lee*, Seoul National University, Se Eun Noh, Myongji University

In kinetic theory of many-body systems, it is well known that the mean-field kinetic equations can effectively describe the large ODE systems. In this talk, we introduce the swarming model on the unit sphere. We study the stability and instability of the incoherent state where all particles are uniformly distributed on the phase space. These phenomena depend on the interplay between the diffusion and coupling strength between the particles.

2010 Mathematics Subject Classification: 92D25, 74A25, 76N10
Key Words and Phrases: Kinetic equations, Lohe model, stability, instability

⋅ 6th-H-17:30 − 18:00 A local sensitivity analysis for the kinetic Kuramoto equation with random inputs (Seung-Yeal Ha, Shi Jin, Jinwook Jung)

하승열((서울대)), Shi Jin((University of Wisconsin–Madison)), 정진욱*((서울대))
Seung-Yeal Ha, Seoul National University, Shi Jin, University of Wisconsin-Madison, Jinwook Jung*, Seoul National University

We conduct a local sensivity analysis for the kinetic Kuramoto equation with random inputs in a large coupling regime. In our proposed random kinetic Kuramoto equation (in short, RKKE), the random inputs are encoded in the coupling strength. For the deterministic case, it is well known that the kinetic Kuramoto equation exhibits asymptotic phase concentration for well-prepared initial data in the large coupling regime. To see a response of the system to the random inputs, we provide propagation of regularity, local-in-time stability estimates for the variations of the random kinetic density function in random parameter space. For identical oscillators with the same natural frequencies, we introduce a Lyapunov functional measuring the phase concentration, and provide a local sensitivity analysis for the functional.

2010 Mathematics Subject Classification: 35Q82, 35Q92, 37H99
Key Words and Phrases: Kuramoto model, local sensitivity analysis, random communication, synchronization, uncertainty quantification
Analysis II

⋅ 4th-C-16:00 − 18:00 Chair: Jinmyoung Seok (Kyonggi University)

⋅ 4th-C-16:00 − 16:30 [CANCELLED] Linear estimates for the dyadic operators on weighted Lebesgue spaces (Daewon Chung)

정대원((계명대))
Daewon Chung, Keimyung University

In this talk, the operator norm on the weighted Lebesgue space $L^{2}(w)$ of some dyadic operators, such as dyadic paraproduct and commutator of the Hilbert transform with a vanishing mean oscillation function $b$, will be discussed. It will be presented that the commutator of the Hilbert transform with a vanishing mean oscillation function $b$ depends linearly on the $A_2$-characteristic of the weight ($[w]_{A_2}$), as opposed to the quadratic dependence known to hold for the commutator of the Hilbert transform with BMO functions. The quadratic dependence of $[w]_{A_2}$ of the commutators with BMO functions is known to be sharp as well as the linear dependence of $[w]_{A_2}$ of the dyadic paraproduct. However, by choosing a function b in the space VMO, we can reduce the dependence to the linear which also known to hold for the Hilbert transform. Then the sharp extrapolation theorem return the following bound in $L^p(w)$, for $w\in A_p$: $$\|[H,b]f\|_{L^p(w)}\leq C(p) [w]_{A_p}^{max\{1,1/(p-1)\}}\|f\|_{L^p(w)}\,.$$

2010 Mathematics Subject Classification: 42B20
Key Words and Phrases: weighted norm estimate, commutator, dyadic paraproduct

⋅ 4th-C-16:30 − 17:00 Evaluation formulas for integrals similar to the conditional Wiener integrals over continuous paths (Dong Hyun Cho)

조동현((경기대))
Dong Hyun Cho, Kyonggi University

Let $(C[0,T],w_{\alpha,\beta})$ denote an analogue of a generalized Wiener space, the space of continuous real-valued functions on the interval $[0,T]$, where $\alpha,\beta:[0,T]\to\mathbb R$ are absolutely continuous functions such that $\beta$ is strictly increasing, and $\varphi$ is an arbitrary finite measure on the Borel class of $\mathbb R$. Define $X_n:C[0,T]\to\mathbb R^n$ by $$X_n(x) = (\int_0^Te_1(s)dx(s),\ldots, \int_0^Te_n(s)dx(s)),$$ where $\{e_1,\ldots,e_n)\}$ is an orthonormal subset of $L_{\alpha,\beta}^2[0,T]$ which is an $L^2$-space with respect to the Lebesgue-Stieltjes measure induced by $\alpha$ and $\beta$. In this talk, we will derive a simple formula for integrals on $C[0,T]$ with the function $X_n$, which generalize the conditional Wiener integrals on $C[0,T]$. As applications of the simple formula, we will evaluate the integrals of various function defined on $C[0,T]$.

2010 Mathematics Subject Classification: 28C20
Key Words and Phrases: analogue of Wiener measure, analogue of Wiener space, conditional Wiener integral, simple formula for conditional Wiener integrals

⋅ 4th-C-17:00 − 17:30 Behavior of a Laplace transform for function space integral (Young Sik Kim)

김영식((한양대))
Young Sik Kim, Hanyang University

We investigate the behavior of the Laplace transform for the function space integral and we investigate the behavior of the convolution under the Laplace transform for the function space integral.

2010 Mathematics Subject Classification: 26D10, 42B35, 42C4, 28C20
Key Words and Phrases: Laplace transform, function space integral

⋅ 4th-C-17:30 − 18:00 Some stability results of functional equation in modular spaces (Hark-Mahn Kim, Hwan Yong Shin)

김학만((충남대)), 신환용*((충남대))
Hark-Mahn Kim, Chungnam National University, Hwan Yong Shin*, Chungnam National University

Ulam type stability is a famous topic in functional Analysis. Recently, many stability results of functional equation have been dealt in convex modular spaces. In this presentation, we consider additive functional equation and quadratic functional equation and investigate the stability of these functional equations in convex modular spaces.

2010 Mathematics Subject Classification: 39B52, 39B74, 47H09
Key Words and Phrases: modular spaces, additive functional equation, quadratic functional equation, Fatou property, $\Delta_2$-condition
Geometry

⋅ 4th-C-16:00 − 17:00   Chair: Jinsung Park (KIAS)

⋅ 4th-C-16:00 − 16:30 The BFK-gluing formula for zeta-determinants and curvature tensors (Yoonweon Lee)

이윤원((인하대))
Yoonweon Lee, Inha University

The coefficients of the asymptotic expansions of the heat trace of a Laplace operator on a compact Riemannian manifold are expressed by curvature tensors including scalar curvatures, Ricci tensor, and principal curvatures, etc. The gluing formula for the zeta-determinants of a Laplace operator was proved by Burghelea, Friedlander and Kappeler. This formula contains a defect constant, which is determined by some geometric data on a small collar neighborhood of a cutting hypersurface. In this talk, we show that this constant is expressed by some curvature tensors including scalar curvatures and principal curvatures of the cutting hypersurface in the ambient manifold, like the case of the heat trace asymptotics.

2010 Mathematics Subject Classification: 58J20
Key Words and Phrases: zeta-determinants, BFK-gluing formula, Dirichlet-to-Neumann operator, heat trace asymptotics, curvature tensors

⋅ 4th-C-16:30 − 17:00 Tropical geometry for super complex tori (Hoil Kim)

김호일((경북대))
Hoil Kim, Kyungpook National University

Tropical geometry is a combinatorial approach for complex variety and nonarchimedian geometry. It is also used to describe the phenomena of mirror symmetry with the relation to log geometry. Abelian variety or complex torus are one of the basic building blocks in geometry and their super versions are needed in mirror symmetry and mathematical physics. In this talk we describe the tropical behaviours of super complex torus(super abelian variety).

2010 Mathematics Subject Classification: 05E, 14K, 14T
Key Words and Phrases: tropical geometry, spherical variety, supertorus, toric variety

⋅ 4th-C-17:00 − 17:20   Chair: Yoonweon Lee (Inha University)

⋅ 4th-C-17:00 − 17:20 Rotationally and birotationally symmetric homothetic solitons for the inverse mean curvature flow (Daehwan Kim, Juncheol Pyo)

김대환*((고등과학원)), 표준철((부산대))
Daehwan Kim*, KIAS, Juncheol Pyo, Pusan National University

The inverse mean curvature flow (IMCF) has been extensively studied not only as a type of geometric flows, but also for its applications to geometric inequalities. The focus is on Homothetic soliton as a special solution of IMCF, which is deformed only homothetically by the flow. We completely classify birotationally symmetric homothetic solitons that are $O(m)\times O(n)$-invariant hypersurfaces in $R^{m+n}$ using the phase-plane analysis, and analyze their asymptotic behavior at infinity using a suitable coordinate transformation. In particular, if $m \geq 2$ and $n = 1$, then the homothetic solitons have rotational symmetry.

2010 Mathematics Subject Classification: 53C44, 37C10, 53A10
Key Words and Phrases: inverse mean curvature flow, homothetic soliton, rotationally and birotationally symmetric hypersurfaces

⋅ 6th-G-13:30 − 14:00   Chair: Jong-Do Park (Kyung Hee University)

⋅ 6th-G-13:30 − 14:00 Involutivity for constrained Pfaffian systems (Chong Kyu Han)

한종규((서울대))
Chong Kyu Han, Seoul National University

Given a set of smooth ($C^{\infty}$) vector fields $X_1, \ldots, X_{\ell} $ that are linearly independent everywhere on a smooth manifold $M^n$ and a smooth map $h: M\rightarrow \mathbb R^p, $ we discuss the problem of deciding the existence of integral manifolds of any dimension $\le n-p, $ that are contained in the zero set of $h$. Let $\theta$ be the Pfaffian system of the independent $1$-forms that annihilate $X$'s. The author presents in this talk a method of prolongation of the ideal ($h, \theta)$, to an involutive system, which proves the existence or non-existence of integral manifolds. The author will present hopefully some applications to affine dynamical control systems, where the map $h$ is the output function of the control system.

2010 Mathematics Subject Classification: 37C10, 57R27, 58A17, 93B05
Key Words and Phrases: Pfaffian system, closed ideal, involutivity, integral manifolds, affine control systems

⋅ 6th-G-14:00 − 15:00   Chair: Jae Hyouk Lee (Ewha Womans University)

⋅ 6th-G-14:00 − 14:30 A one-parameter family of hypersurfaces in the nearly K\"ahler 6-sphere (Jihong Bae, Jeonghyeong Park, Kouei Sekigawa)

배지홍*((공군사관학교)), 박정형((성균관대)), Kouei Sekigawa((Niigata University))
Jihong Bae*, Republic of Korea Air Force Academy, Jeonghyeong Park, Sungkyunkwan University, Kouei Sekigawa, Niigata University

We present two kinds of almost contact metric structures on a one-parameter family of totally umbilical hyperspheres in the nearly K\"ahler unit 6-sphere $S^6$ and introduce explicit examples of the nonexistence of totally umbilical hypersurfaces in the nearly K\"ahler 6-sphere with respect to the naturally induced almost contact metric structure.

2010 Mathematics Subject Classification: 53C25, 53D10
Key Words and Phrases: contact metric manifold, nearly K\"ahler manifold

⋅ 6th-G-14:30 − 15:00 Liouville property of smooth projective symmetric varieties with Picard number one (Shin-Young Kim, Kyeong-Dong Park)

김신영((Institut Fourier)), 박경동*((기초과학연구원))
Shin-Young Kim, Institut Fourier, Kyeong-Dong Park*, IBS-CGP

For a simply connected semisimple complex Lie group $G$ and an involution of $G$, if $H$ is a closed Lie subgroup of $G$ sitting between the invariant subgroup of the given involution and its normalizer, then we say that the homogeneous space $G/H$ is a symmetric space. Vust proved that a normal $G$-variety together with an equivariant open embedding of a symmetric space, called a symmetric variety, is a spherical variety. From the Luna-Vust theory on spherical varieties, embeddings of spherical homogeneous spaces can be classified via colored fans. In 2011, Ruzzi classified smooth projective symmetric varieties with Picard number one by case-by-case study using colored fans. In particular, the such varieties whose restricted root systems are of type $A_2$ can be described as hyperplane sections of the Legendrian manifolds in the geometric Freudenthal-Tits magic square. On the other hand, we say that a uniruled projective manifold has the Liouville property if every local vector field preserving the cone structure associated to the variety of minimal rational tangents (VMRT) can be extended to a global holomorphic vector field on a Zariski open subset. In this talk, we will discuss the Liouville property of a nonhomogeneous smooth projective symmetric variety with Picard number one by considering the automorphism group of this manifold, the Lie algebra of infinitesimal automorphisms of the affine cone of VMRT, and its prolongations.

2010 Mathematics Subject Classification: 53C10, 53C30, 14M27
Key Words and Phrases: Liouville property, smooth symmetric variety, geometric Freudenthal-Tits magic square
Topology

⋅ 4th-C-16:00 − 18:00   Chair: Keonhee Lee (Chungnam National University)

⋅ 4th-C-16:00 − 16:30 Bipartite intrinsically knotted graphs with 23 edges (Hyoungjun Kim, Seungsang Oh, Thomas Mattman)

김형준*((이화여대)), 오승상((고려대)), Thomas Mattman((California State University-Chico))
Hyoungjun Kim*, Ewha Womans University, Seungsang Oh, Korea University, Thomas Mattman, California State University-Chico

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. It is already known that intrinsically knotted graphs have at least 21 edges, and K7 and the 13 graphs obtained from K7 by ∇-Y moves are the only intrinsically knotted graphs with 21 edges. There are exactly two minor minimal bipartite intrinsically knotted graphs with 22 edges. The goal of this talk is to show that there is no minor minimal bipartite intrinsically knotted graphs with 23 edges.

2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: intrinsically knotted graph, spatial graph theory

⋅ 4th-C-16:30 − 17:00 Configuration space, moduli space and 3-fold covering space (Byung Chun Kim, Yongjin Song)

김병천*((인하대)), 송용진((인하대))
Byung Chun Kim*, Inha University, Yongjin Song, Inha University

A function from configuration space to moduli space of surface may induce a homomorphism between their fundamental groups which are braid groups and mapping class groups of surface, respectively. This map $\phi: B_k \rightarrow \Gamma_{g,b}$ is induced by 3-fold branched covering over a disk with some branch points. In this talk we give a description of this map and show that it is injective by Birman-Hilden theory. This gives us a new interesting non-geometric embedding of braid group into mapping class group. On the other hand, we show that the map on the level of classifying spaces of groups is compatible with the action of little 2-cube operad so that it induces a trivial homomorphism between stable homology group of braid groups and that of mapping class groups(Harer conjecture). We also show how the lift $\tilde{\beta_i}$ acts on the fundamental group of the surface and through this we prove that $\tilde{\beta_i}$ equals the product of two inverse Dehn twists.

2010 Mathematics Subject Classification: 57M50, 57M12, 57M50
Key Words and Phrases: configuration space, moduli space, branched coverings, operad

⋅ 4th-C-17:00 − 17:30 Torsions in the cohomology of real toric manifold (Suyoung Choi)

최수영((아주대))
Suyoung Choi, Ajou University

For each toric variety $X$, there is a canonical involution induced by a conjugation. The set of its fixed points, denoted by $X^{\mathbb R}$ is called a real toric variety. If $X$ is complete and non-singular, then $X$ and $X^{\mathbb R}$ are called a \emph{toric manifold} and a \emph{real toric manifold}, respectively. The integral cohomology $H^\ast(X)$ of toric manifold $X$ is well-known. In particular, it has no torsion element. Nevertheless, only little is known about the integral cohomology of $X^{\mathbb R}$. Recently, the speaker and his collaborators have developed the fundamental framework of real toric spaces. In this talk, we discuss torsion elements in $H^\ast(X^{\mathbb R})$. We show that any torsion can appear in $H^\ast(X^{\mathbb R})$. Furthermore, we find the necessary and sufficient condition for real toric manifolds to have a torsion-free cohomology group.

2010 Mathematics Subject Classification: 57N65
Key Words and Phrases: real toric manifold, torsion, cohomology, toric topology

⋅ 4th-C-17:30 − 18:00 Free actions of finite nonabelian groups on the 3-dimensional nilmanifold with the first homology $\mathbb Z^2 \oplus \mathbb Z_2$ (Joonkook Shin, Daehwan Koo, Mi Na Han)

신준국*((충남대)), 구대환((대전과학고)), 한미나((천안월봉중))
Joonkook Shin*, Chungnam National University, Daehwan Koo, Daejeon Science High School, Mi Na Han, Cheonan Wolbong Middle School

Free actions of finite groups on the $3$-dimensional nilmanifold have been studied step by step by several researchers. First, H. Chu and J. Shin showed that a finite group acting freely on the standard $3$-dimensional nilmanifold $\mathcal N_1$ with the first homology $\mathbb Z^2$ is cyclic ({\em Topology Appl.} {\bf 144} (2004), 255--270). Next, free actions of finite abelian groups on the $3$-dimensional nilmanifold $\mathcal N_p$ with the first homology $\mathbb Z^2\oplus\mathbb Z_p$ were classified by D. Choi and J. Shin ({\em J. Korean Math. Soc.} {\bf 42(4)} (2005), 795--826). In this article, we focus on the free actions of finite nonabelian groups on $\mathcal N_2$ and clarify what those groups are. We study the such actions for $15$ cases of the $3$-dimensional almost Bieberbach groups, and show that there exist three kinds of nonabelian free actions in $\pi_1$ or $\pi_4$, two in $\pi_2$ or $\pi_{5,i}(i=1,2,3)$, and one in the other cases.

2010 Mathematics Subject Classification: 57S25, 57M05
Key Words and Phrases: affine conjugacy, almost Bieberbach group

⋅ 5th-D-09:00 − 10:00   Chair: Seungsang Oh (Korea University)

⋅ 5th-D-09:00 − 09:30 Length of chains composed by certain monoids of self maps (Howon Choi, Keeyoung Lee)

최호원*((고려대)), 이기영((고려대))
Howon Choi*, Korea University, Keeyoung Lee, Korea University

For a based CW-complex $X$, $\mathcal{A}_\sharp^n(X)$ is the submonoid of $[X,X]$ which consists of all homotopy classes of self maps of $X$ that induce an automorphism on $\pi_k(X)$ for all $0\leq k \leq n$. Since, for $m<n$, $\mathcal{A}_{\sharp}^ n(X)\subseteq \mathcal{A}_{\sharp}^m(X)$, there is a chain by inclusions: $\mathcal{E}(X)\subseteq\mathcal{A}_{\sharp}^{\infty}(X)\subseteq \cdots \subseteq \mathcal{A}_{\sharp}^1(X)\subseteq \mathcal{A}_{\sharp}^0(X)=[X,X].$ In this paper, we study the number of strict inclusions in this chain for given connected CW-complex. We call this number the self length of a given space. We prove that the number is a homotopy invariant and indicate a close connection with the self closeness number which is the minimum number $n$ such that $\mathcal{E}(X)=\mathcal{A}_{\sharp}^n(X)$. Moreover, we determine the self-lengths of several spaces and give the lower bounds or upper bounds of the self-lengths of some spaces.

2010 Mathematics Subject Classification: 55P10, 55P20, 55Q05, 55Q20
Key Words and Phrases: homotopy equivalence, self-closeness number, self-length

⋅ 5th-D-09:30 − 10:00 Self-homotopy equivalences of Moore spaces depending on cohomotopy groups (Hyung Seok Oh, Kee Young Lee, Ho Won Choi)

오형석*((고려대)), 이기영((고려대)), 최호원((고려대))
Hyung Seok Oh*, Korea university, Kee Young Lee, Korea university, Ho Won Choi, Korea university

Given a topological space $X$ and a non-negative integer $k$, $\mathcal{E}_{k}^{\sharp}(X)$ is the set of all self-homotopy equivalences of $X$ that do not change maps from $X$ to an $t$-sphere $S^{t}$ homotopically by the composition for all $t \geq k$. This set is a subgroup of the self-homotopy equivalence group $\mathcal{E}(X)$. We find certain homotopic tools for computations of $\mathcal{E}_{k}^{\sharp}(X)$. Using these results, we determine $\mathcal{E}_{k}^{\sharp}(M(G,n))$ for $k\geq n$, where $M(G, n)$ is a Moore space type of $(G, n)$ for a finitely generated abelian group $G$.

2010 Mathematics Subject Classification: 55P10, 55Q05, 55Q55
Key Words and Phrases: self-homotopy equivalence, cohomotopy group, Moore space, co-Moore space

⋅ 6th-F-09:00 − 09:50   Chair: Suyoung Choi (Ajou University)

⋅ 6th-F-09:00 − 09:30 Periodic shadowing property for continuous flows (Namjip Koo, Nyamdavaa Tsegmid)

구남집((충남대)), Nyamdavaa Tsegmid*((충남대))
Namjip Koo, Chungnam National University, Nyamdavaa Tsegmid*, Chungnam National University

P. Walters proved that an expansive homeomorphism $T:X\to X$ of a compact metric space $(X,d)$ with the pseudo orbit tracing property(POTP) is topologically stable. Then, R. F. Thomas obtained some versions of flows concerning the Walters' results and showed that the suspension flow $\varphi$ on $(Y,\rho)$ of a homeomorphism $T$ on $(X,d)$ under a continuous map $f: X \to \mathbb{R} > 0$ has POTP if and only if $T$ has POTP. In this talk we introduce some relations between the periodic shadowing property and POTP for continuous flows on a compact metric space and discuss the Thomas' results for flows via the notion of the periodic shadowing property.

2010 Mathematics Subject Classification: 37C50, 37C27, 37C55
Key Words and Phrases: periodic shadowing property, flow, suspension flow

⋅ 6th-F-09:30 − 09:50 Dynamics on inverse limit systems (Nakyung Lee, Hahng-Yun Chu)

이나경*((충남대)), 주항연((충남대))
Nakyung Lee*, Chungnam National University, Hahng-Yun Chu, Chungnam National University

In this talk, we discuss on dynamical properties of inverse limit systems which consist of two cross bonding function systems. We focus on horizontal shift maps on the systems and investigate expansiveness, shadowing property and monotonicity of the maps.

2010 Mathematics Subject Classification: 54H20, 37B50
Key Words and Phrases: shift map, expansivity, shadowing property, monotonicity

⋅ 6th-G-13:30 − 15:30   Chair: Sang Youl Lee (Pusan National University)

⋅ 6th-G-13:30 − 14:00 A generalization of the rectangle condition (Bo-hyun Kwon)

권보현((고려대))
Bo-hyun Kwon, Korea University

In this talk, we define the \textit{connecting rectangle condition} to check whether or not a Heegaard splitting is strongly irreducible which is a variation of the rectangle condition by Casson and Gordon. We define a \textit{general version} of the rectangle condition. Moreover, with a similar condition defined on an $n$-bridge decomposition, we can check whether or not the Hempel distance of an $n$-bridge decomposition is greater than or equal to two.

2010 Mathematics Subject Classification: 57M27
Key Words and Phrases: connecting rectangle condition, strongly irreducible, bridge decomposition

⋅ 6th-G-14:00 − 14:30 Remarks on the Liechti-Strenner's examples having small dilatations (Ji-Young Ham, Joongul Lee)

함지영*((중앙대)), 이준걸((홍익대))
Ji-Young Ham*, Chung-Ang University, Joongul Lee, Hongik University

We show that Liechti-Strenner's examples for the closed non-orientable surfaces of even genus in ``Minimal pseudo-Anosov stretch factors on nonorientable surfaces" are minimal among the monic polynomials with negative coefficients. The same is true for the orientation-reversing ones for the closed orientable surfaces of odd genus.

2010 Mathematics Subject Classification: 37E30, 37B40, 57M60
Key Words and Phrases: minimal dilatation, nonorientable surface, Liechti-Strenner, pseudo-Anosov stretch factors

⋅ 6th-G-14:30 − 15:00 Stability properties of $p$-adic dynamical systems (Gansukh Tumur, Namjip Koo)

Gansukh Tumur*((충남대)), 구남집((충남대))
Gansukh Tumur*, Chungnam National University, Namjip Koo, Chungnam National University

Let $p$ be a fixed number and $S_p$ be the set of all $p$-sequences. In this talk we briefly recall some algebraical and topological properties of the $p$-sequence set $S_p$. Then we will discuss the shadowing property and expansiveness of $p$-adic discrete dynamical systems on $S_p$ given by linear maps and shift maps.

2010 Mathematics Subject Classification: 37P99
Key Words and Phrases: $p$-adic dynamical system, $p$-sequence set

⋅ 6th-G-15:00 − 15:30 Decomposition of factor maps (Jisang Yoo)

유지상((성균관대))
Jisang Yoo, Sungkyunkwan University

Given a factor map $\pi:X\to Y$ between topological dynamical systems $X$ and $Y$, Glasner and Weiss introduced the notion of relative Pinsker factor $Z$ in order to decompose the map $X\to Y$ into a completely positive entropy $X\to Z$ and zero entropy $Z\to Y$. Examples of relative Pinsker factors are well known when $Y$ is trivial. But when $Y$ is not trivial, no examples are known in the case of symbolic dynamical systems. In this talk, we define a closely related notion of class decomposition of factor maps between symbolic dynamical systems, exploiting recent results from degree theory of symbolic dynamics. We show that class decomposition is almost an example of relative Pinsker factor.

2010 Mathematics Subject Classification: 37B10, 37B40
Key Words and Phrases: Pinsker factor, class degree, symbolic dynamics

⋅ 6th-H-16:00 − 18:00   Chair: Dae-Woong Lee (Chonbuk National University)

⋅ 6th-H-16:00 − 16:30 Gromov-Hausdorff dynamical systems (C. A. Morales)

C. A. Morales, Federal University of Rio de Janeiro

We explain how the Gromov-Hausdorff space theory can be applied to dynamical systems.

2010 Mathematics Subject Classification: 54H20
Key Words and Phrases: Gromov Hausdorff space, dynamical system, homeomorphism

⋅ 6th-H-16:30 − 17:00 Symbolic topological dynamics in the circle (Ju Mi Oh, C. A. Morales)

오주미*((성균관대)), C. A. Morales((Federal University of Rio de Janeiro))
Ju Mi Oh*, Sungkyunkwan University, C. A. Morales, Federal University of Rio de Janeiro

In this talk, we explain how dynamical systems with generating partitions are symbolically expansive namely symbolic counterparts of the expansive ones. Similar ideas allow the notions of symbolic equicontinuity, symbolic distality, symbolic N-expansivity and symbolic shadowing property. We analyze dynamical systems with these properties in the circle.

2010 Mathematics Subject Classification: 37B05
Key Words and Phrases: symbolically expansive, symbolic shadowing, metric space

⋅ 6th-H-17:00 − 17:30 Weak shadowing property for robust diffeomorphisms (Manseob Lee)

이만섭((목원대))
Manseob Lee, Mokwon University

Let $M$ be a compact $n(\geq2)$-dimensional manifold without boundary, and let ${\rm Diff}(M)$ be the space of $C^1$ diffeomorphisms of $M$. We show that if a diffeomorphism $f$ has the $C^1$ robustly weak shadowing property on a nontrivial transitive set $\Lambda\subset M$ then $\Lambda$ is hyperbolic for $f$. We say that $f$ is {\it tame} if there is a $C^1$ neighborhood $\mathcal{U}(f)$ such that for any $g\in\mathcal{U}(f)$, $g$ has only finitely many chain recurrence class. Gan suggested that if a diffeomorphism $f$ has the $C^1$ robustly weak shadowing property then is it tame? It is clear that if a diffeomorphism $f$ is Anosov then it is tame. The result is a partial answer of Gan's conjecture (see [1]). \begin{thebibliography}{1} \bibitem{Y} D. Yang, {\it Stably weakly shadowing transitive sets and dominated splttings}, Proc. Amer. Math. Soc. {\bf 269} (2011), 801--807. \end{thebibliography}

2010 Mathematics Subject Classification: 37C50, 37D20
Key Words and Phrases: shadowing, weak shadowing, transitive, local star, hyperbolic

⋅ 6th-H-17:30 − 18:00 Continuous dynamical systems with canonical coordinates (Seunghee Lee)

이승희((국가수리과학연구소))
Seunghee Lee, NIMS

Canonical coordinates are very useful for ergodic theory, entropy calculations and topological dynamics. Since canonical coordinates move around with a point, we may extend certain notions which are valid for fixed or periodic points to this setting. Very recently, discrete dynamical systems with canonical coordinates are deal with robust dynamic property in [2] which has used R. Mane's result in [1]. In this talk, we discuss that a continuous dynamical system with canonical coordinates has periodic orbits shadowing every recurrent orbit. \begin{thebibliography}{9} \bibitem{1} R. Mane, {\it Contributions to the conjecture stability}, Topology {\bf 17} (1978), 383--396. \bibitem{2} M. Lee and J. Park, {\it Expansive transitive sets for robust and generic diffeomorphisms}, Dynamical Systems {\bf 33} (2017), 1--13. \end{thebibliography}

2010 Mathematics Subject Classification: 37C25, 37C75
Key Words and Phrases: canonical coordinate, shadowing, periodicity
Probability and Statistics

⋅ 4th-C-16:00 − 17:30 Chair: Hyun Jae Yoo (Hankyong University)

⋅ 4th-C-16:00 − 16:30 Heat kernel estimates for symmetric jump processes with general mixed polynomial growths (Panki Kim)

김판기((서울대))
Panki Kim, Seoul National University

In this talk, we discuss transition densities of pure jump symmetric Markov processes whose jumping kernels are comparable to radially symmetric functions with general mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of transition densities (heat kernel estimates) for such processes. This is a joint work with Joohak Bae, Jaehoon Kang and Jaehun Lee.

2010 Mathematics Subject Classification: 60-xx
Key Words and Phrases: transition density, jump processes, Markov processes

⋅ 4th-C-16:30 − 16:50 local law and tracy-widom limit for sparse sample covariance matrices (JongYun Hwang, JiOon Lee, Kevin Schnelli)

황종연*((한국과학기술원)), 이지운((한국과학기술원)), Kevin Schnelli((KTH Royal Institute of Technology))
JongYun Hwang*, KAIST, JiOon Lee, KAIST, Kevin Schnelli, KTH Royal Institute of Technology

We consider spectral properties of sparse sample covariance matrices, the class of random matrices including biadjacency matrices of the bipartite Erd\H{o}s--R\'enyi graph model. We prove a local law for the eigenvalue density up to the upper spectral edge. Under a suitable condition on the sparsity, we also prove that the rescaled, shifted extremal eigenvalue exhibit GOE Tracy-Widom law on the deterministic shift of the spectral edge.

2010 Mathematics Subject Classification: 60B20, 62H10
Key Words and Phrases: high dimensional sample covariance matrices, local law, Tracy-Widom distribution

⋅ 4th-C-17:00 − 17:30 Graph-based P\'olya urns on countable networks (Christian Hirsch, Mark Holmes, Victor Kleptsyn)

Christian Hirsch*, Aalborg University, Mark Holmes, The University of Melbourne, Victor Kleptsyn, University of Rennes 1

We examine a reinforcement model on the edges of a countably infinite network. The edge weights evolve according to a stochastic process subject to the following dynamics. First, vertices fire at times given by a Poisson point process. Second, if a vertex fires at a point in time, then one of the incident edges is selected with a probability proportional to the current weight to a power $\alpha>1$. Third, the weight of the selected edge is increased by 1. We analyze the connectivity structure of the family of edges used a positive proportion of time. This talk is based on joint work with Mark Holmes and Victor Kleptsyn.

2010 Mathematics Subject Classification: 60K35, 82B43, 82C43
Key Words and Phrases: percolation, reinforcement model, P\'olya urns, stochastic approximation algorithm,
Applied Mathematics

⋅ 4th-C-16:00 − 17:30 Chair: Shinuk Kim (Sangmyung University)

⋅ 4th-C-16:00 − 16:30 Robust coexistence as a global attractor in the spatial dynamics of cyclic competition (Junpyo Park)

박준표((울산과학기술원))
Junpyo Park, UNIST

In the past decade, there have been many efforts to understand species interplay with biodiversity in rock-paper-scissors (RPS) games within macro and microscopic levels. In this direction, mobility and intraspecific competition have been found to be the main factors promoting coexistence in spatially extended systems. In this paper, we explore the relevant effect of asymmetric competitions coupled with mobility on the coexistence of cyclically competing species. By examining the coexistence probability, we have found that mobility can facilitate coexistence in the limited cases of asymmetric competition and can be well predicted by the basin structure of the deterministic system. In addition, it is found that mobility can have beneficial and harmful effects on coexistence when all competitions occur asymmetrically. We also found that the coexistence in spatial dynamics of RPS game ultimately becomes a global attractor.

2010 Mathematics Subject Classification: 37N25, 91A25, 92D25
Key Words and Phrases: Rock-paper-scissors game, mobility, spatial dynamics, global attractor

⋅ 4th-C-16:30 − 17:00 A generalized Cucker-Smale model on a Riemannian manifold and its emergent dynamics (Seung-Yeal Ha, Doheon Kim, Franz Wilhelm Schl\"oder)

하승열((서울대)), 김도헌*((서울대)), Franz Wilhelm Schl\"oder((University of Milano-Bicocca))
Seung-Yeal Ha, Seoul National University, Doheon Kim*, Seoul National University, Franz Wilhelm Schl\"oder, University of Milano-Bicocca

We present a generalized Cucker-Smale(C-S) model on a Riemannian manifold and study its emergent dynamics. For the constant mean curvature manifolds such as the unit sphere in $\mathbb S^2$ and hyperbolic plane $\mathbb H^2$, we explicitly present a sufficient framework leading to the asymptotic flocking for the proposed generalized model.

2010 Mathematics Subject Classification: 34C40, 34D05, 70F99
Key Words and Phrases: Cucker-Smale model, flocking, Riemannian manifold

⋅ 4th-C-17:00 − 17:30 Computational models for clinical information (Shinuk Kim)

김신욱((상명대))
Shinuk Kim, Sangmyung University

We introduce a computational model based on clinical datasets.

2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: computational models cancer datasets
Mathematical Education

⋅ 6th-F-09:00 − 09:30 Chair: Eunmi Choi (Hannam University)

⋅ 6th-F-09:00 − 09:30 Interactive Discrete Mathematics Laboratory using SageMath (Sang-Gu Lee, Jae Hwa Lee)

이상구*((성균관대)), 이재화((성균관대))
Sang-Gu Lee*, Sungkyunkwan University, Jae Hwa Lee, Sungkyunkwan University

This study deals with the content that was developed by the authors and the utilization of the `Interactive Discrete Mathematics Laboratory (IDML).' IDML provides online review lectures as well as a wide range of examples and exercises from the curriculum of discrete mathematics courses. The lectures come with pre-coded python-based SageMath cells through which students can run and modify the code directly from this free laboratory. IDML is accessible via mobile devices so that the students can use it anywhere, anytime for maximum learning effectiveness and achievement. IDML would be an ideal tool for the effective learning and teaching of discrete mathematics, which combines theory and practice.

2010 Mathematics Subject Classification: 97U70, 97K20, 97K30
Key Words and Phrases: Discrete Mathematics, Math Cyber Laboratory, SageMath

⋅ 6th-G-13:30 − 14:30 Chair: Sunsook Noh (Ewha Womans University)

⋅ 6th-G-13:30 − 14:00 Some views of mathematicians with focus on pedagogics (Seong-A Kim)

김성아((동국대))
Seong-A Kim, Dongguk University (Gyeongju)

I will share some observations on teaching mathematics in the university from a view of mathematician with focus on pedagogics who have also worked on mathematics education. As a mathematician, I had finished program of doctoral degree in math education and have published articles in the area too. I have taught both courses in math and math education since 2006. There has been noticeably different opinions between mathematician and math educators in teaching college math[ref. 'Math War']. The present period asks even math instructors to have concerns on pedagogics as well as mathematics knowledge. Through this talk I hope the audience open and share their experiences and knowledge in teaching college math.

2010 Mathematics Subject Classification: 97A99
Key Words and Phrases: math, math education, pedagogics

⋅ 6th-G-14:00 − 14:30 Pre-service teachers' understanding about geometric interpretation of the derivative of complex-valued functions (Jinhee Park, Ohnam Kwon)

박진희*((서울대)), 권오남((서울대))
Jinhee Park*, Seoul National University, Ohnam Kwon, Seoul National University

Differentiation is an important tool that helps to understand graphs of real-valued functions. Using the derivative of functions, we can draw graphs of real-valued functions on a Cartesian plane. On the other hand, it is difficult to figure out the shape of the graph of a complex-valued function using its derivatives. Nonetheless, the fact that a differentiable real-valued function has a property called local linearity, which means that the graph looks straight when you view it under a microscope, can be applied to any differentiable (analytic) complex-valued functions. Furthermore, since a linear complex-valued function expresses rotation, dilation, and translation, we can locally describe the derivative of a complex-valued function as the rotation and dilation of an image with respect to its pre-image. In this study, the researcher examined eight pre-service teachers' understanding about geometric interpretation of the derivative of complex-valued functions. As a result, they can develop their understanding using analogical reasoning from the property of real-valued functions with the aid of GeoGebra.

2010 Mathematics Subject Classification: 97B50
Key Words and Phrases: complex-valued function, derivative, local linearity
Discrete Mathematics

⋅ 6th-F-09:00 − 10:00 Chair: Sangwook Kim (Chonnam National University)

⋅ 6th-F-09:00 − 09:30 Quasi-Sturmian colorings on regular trees (Dong Han Kim, Seul Bee Lee, Seonhee Lim, Deokwon Sim)

김동한((동국대)), 이슬비*((서울대)), 임선희((서울대)), 심덕원((서울대))
Dong Han Kim, Dongguk University, Seul Bee Lee*, Seoul National University, Seonhee Lim, Seoul National University, Deokwon Sim, Seoul National University

Factor complexity (or subword complexity) of a word is a map assigning a positive integer $n$ to the number of distinct factors of length $n$ appearing in the word. Factor complexity is a classical invariant which measures the disorder of words. Kim and Lim generalized it to factor complexity of a vertex coloring of a regular tree. It assigns $n$ to the number of distinct colorings on balls of radius $n$ which occurs in the coloring. Sturmian words are infinite words with the minimal unbounded factor complexity $n+1$. They are related to irrational rotations and cutting sequences on a square billiard. Quasi-Sturmian words, which are words with factor complexity eventually $n+c$, share many properties with Sturmian words. Kim and Lim defined the Sturmian colorings analogously. They proved an induction algorithm for the Sturmian colorings. In the same sense, we defined the quasi-Sturmian colorings. In this talk, we will discuss induction algorithm for the quasi-Sturmian colorings. Then, we will characterize the quasi-Sturmian colorings by their quotient graphs and recurrence functions. This is joint work with Dong Han Kim, Seonhee Lim and Deokwon Sim.

2010 Mathematics Subject Classification: 05C15
Key Words and Phrases: trees, quasi-Sturmian, colorings, factor complexity, induction algorithm

⋅ 6th-F-09:30 − 10:00 3-dynamic coloring of planar triangulations (Yoshihiro Asayama, Yuki Kawasaki, Seog-Jin Kim, Atsuhiro Nakamoto, Kenta Ozeki)

Yoshihiro Asayama((Yokohama National University)), Yuki Kawasaki((Yokohama National University)), 김석진*((건국대)), Atsuhiro Nakamoto((Yokohama National University)), Kenta Ozeki((Yokohama National University))
Yoshihiro Asayama, Yokohama National University, Yuki Kawasaki, Yokohama National University, Seog-Jin Kim*, Konkuk University, Atsuhiro Nakamoto, Yokohama National University, Kenta Ozeki, Yokohama National University

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring $\phi$ such that for any vertex $v$, $v$ has at least $\min\{r,\deg_G(v) \}$ distinct colors in $N_G(v)$. The {\em $r$-dynamic chromatic number} $\chi_r^d(G)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The {\em list $r$-dynamic chromatic number} of a graph $G$ is denoted by $ch_r^d(G)$. Loeb, Mahoney, Reiniger and Wise (2018) showed that if $G$ is a planar graph, then $\chi_3^d(G) \leq 10$, and there is a planar graph $G$ with $\chi_3^d(G) = 7$. Thus finding an optimal upper bound on $\chi_3^d(G)$ for a planar graph $G$ is a natural interesting problem. In this paper, we show that $\chi_r^d(G) \leq 5$ if $G$ is a planar triangulation. The upper bound is sharp.

2010 Mathematics Subject Classification: 05C15
Key Words and Phrases: graph coloring, dynamic coloring, planar graph

⋅ 6th-G-13:30 − 15:30 Chair: Seog-Jin Kim (Konkuk University)

⋅ 6th-G-13:30 − 14:00 On complexity of cyclic coverings of graphs (Young Soo Kwon, Alexander Mednykh, Ilya Mednykh)

권영수*((영남대)), Alexander Mednykh((Norvosivirsk State University)), Ilya Mednykh((Nor\-vosivirsk State University))
Young Soo Kwon*, Yeungnam University, Alexander Mednykh, Norvosivirsk State University, Ilya Mednykh, Norvosivirsk State University

By complexity of a finite graph we mean the number of spanning trees in the graph. The aim of the present talk is to give a new approach for counting complexity of cyclic coverings of a graph.

2010 Mathematics Subject Classification: 05C50
Key Words and Phrases: complexity of graphs, graph coverings, Laplacian matrix

⋅ 6th-G-14:00 − 14:30 Refined enumeration of singular overpartitions (Seunghyun Seo, Ae Ja Yee)

서승현*((강원대)), 이애자((The Pennsylvania State University))
Seunghyun Seo*, Kangwon National University, Ae Ja Yee, The Pennsylvania State University

Singular overpartitions, which are defined by George Andrews, are overpartitions whose Frobenius symbols have at most one overlined entry in each row. In his paper, Andrews found interesting combinatorial and arithmetic properties of $(k,i)$-singular overpartitions, which are singular overpartitions with some restrictions subject to $k$ and $i$. In this talk, we present a combinatorial proof of a refinement of Andrews's result using the Dyson map and the Wright map.

2010 Mathematics Subject Classification: 05A17, 11P81
Key Words and Phrases: partitions, overpartitions, Frobenius symbols, singular overpartitions, Dyson's map, Wright's map

⋅ 6th-G-14:30 − 15:00 Facial structures of lattice path matroid polytopes (Sangwook Kim)

김상욱((전남대))
Sangwook Kim, Chonnam National University

For a matroid $M$ on $[n]:= \{1, 2, \dots, n \}$, a matroid base polytope $\mathcal{P}(M)$ is the polytope in $\mathbb{R}^n$ whose vertices are the incidence vectors of the bases of $M$. A lattice path matroid is a transversal matroid corresponding to a pair of lattice paths having common end points. The combinatorial and structural properties of lattice path matroids are studied by Bonin et al. In this talk, we study the facial structure of a matroid base polytope which corresponds a lattice path matroid

2010 Mathematics Subject Classification: 52B40
Key Words and Phrases: matroid base polytope, lattice path matroid, face poset

⋅ 6th-G-15:00 − 15:30 On clique adjacency polynomials (Jongyook Park)

박종육((원광대))
Jongyook Park, Wonkwang University

In this talk, we will study clique adjacency polynomials. And we will use these polynomials to obtain a bound on cliques, we call this bound as the clique adjacency bound. We will compare this bound with the Hoffman bound for the class of edge-regular graphs.

2010 Mathematics Subject Classification: 05E30
Key Words and Phrases: clique adjacency polynomials, clique adjacency bounds, the Hoffman bound