Invited Speakers

◎ Plenary Lecture (10.25)


오용근  Oh, Yong-Geun

• 기초과학연구원-기하학수리물리연구단 단장
   Director, Center for Geometry and Physics, The Institute for Basic Science
• 포항공과대학교 수학과 교수
   Distinguished Visiting Professor, Department of Mathematics, POSTECH
• 미국 위스콘신대학교 수학과 교수
   Professor, Department of Mathematics, University of Wisconsin, Madison

Title: Hamilton-Jacobi equation and continuous Hamiltonian
       dynamics
In this talk, I will explain a Floer theoretic construction of solution of Hamilton-Jacobi equation and its relation to continous Hamiltonian dynamics.  I will also indicate how the study is related to problems in area-preserving dynamical systems in 2 dimension.


◎ Public Lecture (10.24)

 
황준묵  Hwang, Jun-Muk

• 고등과학원 수학부 교수   
   Professor, Department of Mathematics, Korea Institute for Advanced Study
   (KIAS)
 
Title: 세계수학자대회 이야기 
2014년 8월 서울에서 세계수학자대회(ICM)가 열립니다.  이 행사를 개최하는 것은 대한민국 수학자 모두의 영광이고 우리 국민 모두가 자랑스러워할 일입니다. 대회에 대한 공식적 안내와 자료는 국제수학연맹(IMU) 홈페이지와 서울 세계수학자대회 홈페이지 및 조직위원회에서 주관하는 각종 홍보와 언론보도를 통해서 얻으실 수 있습니다. 이 강연에서는 강연자의 주관적 견해와 개인적 경험 및 충분히 검증되지 않은 자료를 바탕으로 세계수학자대회에 대해 지극히 비공식적이고 다소 편향된 소개를 해드리고자 합니다.

◎ Invited Lectures (10.25~10.26)

 
[Algebra]  김병한  Kim, Byunghan

• 연세대학교 수학과 교수
   Professor, Department of Mathematics, Yonsei University
• 2013년도 대한수학회상 봄 논문상 수상자
   The winner of 2013 Excellent Research Paper Award

Title: A group configuration theorem in model theory
Group configuration is historically an important issue/problem in model theory, a major branch of mathematical logic. In 1970s B. Zilber resolved the group configuration problem when a structure is stable and $\aleph_0$-categorical. In 80s E. Hrushovski removed $\aleph_0$-categoricity which is a notable progress and later the technique is used in his solution of Mordell-Lang conjecture in any characteristic. In mid 90s, due to the author’s work, the scope of stability theory (studying stable structures) is enlarged to simplicity theory dealing more structures such as the random graph and pseudo-finite fields. In this talk the author’s recent work on solving more general group action configuration problem in simplicity theory and its applications will be surveyed.

[Algebra]  오영탁  Oh, Young-Tak

• 서강대학교 수학과 교수
   Professor, Department of Mathematics, Sogang University
• 2013년도 대한수학회상 봄 논문상 수상자
   The winner of 2013 Excellent Research Paper Award

Title: Classification and decomposition of the Witt-Burnside ring and Burnside ring of a profinite group
The Witt-Burnside ring functor attached to a profinite group $G$ was introduced by Dress and Siebeneicher as a group-theoretical generalization of the classical $p$-typical Witt vectors of Teichm\"{u}ller and Witt and the big Witt vectors of Cartier. As its name implies, it has intimate relation with the Burnside ring of $G$. In this talk, I will explain a few fundamental properties of Witt-burnside rings and  Burnside rings. Recent developments in this area together with connections to other areas will be dealt with, too.

[Analysis]  계승혁  Kye, Seung-Hyeok

• 서울대학교 수리과학부 교수
   Professor, Department of Mathematical Sciences, Seoul National University
• 2012년도 대한수학회상 학술상 수상자
   The winner of 2012 KMS Academic Achievement Award

Title: Geometry for separable states
Distinguishing entanglement from separability is one of the most important question in the theory of quantum entanglement, which is a very useful notion in quantum information thoery as well as quantum physics. In this talk, we compare the local and global geometries for separable states and PPT states for this purpose.

[Analysis]  진범자  Jin, Bum Ja

• 목포대학교 수학교육과 교수
   Professor, Department of Mathematics Education, Mokpo National University

Title: On the regularity of weak solutions to the motion of the degenerate power-law fluids
Let $\Omega$ be a bounded domain in ${\mathbb R}^n, n=2,3$.
We consider the steady and unsteady motion of a fluid described by the systems
\begin{equation}
\label{e1}(u\cdot \nabla )u -\mbox{div}S(Du)+\nabla p =f,\ \mbox{div}u=0 \mbox{ in }\Omega,\ u|_{\partial \Omega}=0\end{equation}
and
\begin{equation}
\label{e2}
u_t+(u\cdot \nabla )u -\mbox{div}S(Du)+\nabla p =0,\ \mbox{div}u=0 \mbox{ in }\Omega\times (0,T),\
u|_{\partial \Omega}=0, u|_{t=0}=a,
\end{equation}
respectively, where 
\begin{equation}
\label{e3}
S(Du)=|Du|^{q-2}Du.
\end{equation}
%Observe that w
When $q=2$, the system becomes (incompressible)Navier-Stokes equations, and the well known theories on the linear (partial differential)operator could be applied to study the regularity of weak solutions.
When $q\neq 2$, the structure of stress tensor is no more linear, whence we cannot apply any known linear operator theory.
To study regularity of weak solutions it is natural to use variational approach such as difference quotient scheme.
Since variational approach is based on the integrability of the convection term
$(u\cdot \nabla v)\cdot w$ for $u,v,w\in W^{1,q}(\Omega)$, it restricts the range of $q$ to $q>\frac{3n}{n+2}$.
The most critical case is that $2<q<3,\ n=3$ for the steady problem, and $3>q>\frac{9}{5}, n=3$ and $2>q>\frac{3}{2}, n=2$ for the unsteady problem.

In this talk, I present two recent works on those cases, which were obtained by myself with other coworkers.

[Geometry]  Filippo Morabito

• 카이스트 수리과학과 교수
   Professor, Korea Advanced Institute of Science and Technology (KAIST)

Title: Delaunay type domains for an overdetermined elliptic problem
We construct new domains in $S^n \times R$ and $H^n \times R$ which are critical points
for the map $D \to \lambda_1(D),$ where $\lambda_1$ is the first eigenvalue of the Laplace-Beltrami operator and $D$ is a domain, by the technique of the bifurcation.
Such domains can be characterized as domains for which an ovedetermined boundary value problem admits a positive solution.

[Special Invited Lecture - Topology]  Dale Rolfsen

• Professor, Department of Mathematics, The University of British Columbia

Title: Groups of self-homeomorphisms of manifolds
In 1920, Kerekjarto proved that the group of homeomorphisms of the 2-dimensional disk which are pointwise fixed on the boundary has no elements of finite order.  In joint work with Danny Calegari, I will discuss why the same is true in higher dimensions, at least for the subgroup of piecewise linear homeomorphism.   More general results on algebraic properties of groups of PL self-homeomorphisms will be proved.

[Topology]  김상현  Kim, Sang-hyun

• 카이스트 수리과학과 교수
   Professor, Korea Advanced Institute of Science and Technology (KAIST)
• 2012년도 상산 젊은수학자상 수상자
   The winner of 2012 Sangsan Prize for young Mathematicians

Title: A curve complex for a right-angled Artin group
For each right-angled Artin group G, we associate a quasi-tree T. Combinatorially, this quasi-tree is characterized as the ``iterated double'' of the defining graph of G. Algebraically, T roughly encodes the isomorphism types of right-angled Artin subgroups of G. And geometrically, we observe that G acts on T acylindrically and that T plays an analogous role to a curve complex for a mapping class group. Corollaries include embeddability results between RAAGs, and also from RAAGs to mapping class groups.

[Probability and Statistics]  Gerald Trutnau

• 서울대학교 수리과학부 교수
   Professor, Department of Mathematical Sciences, Seoul National University
 
Title: About the stochastic regularity of distorted Brownian
Let $\Omega\subset \mathbb{R}^d$ be an open or closed domain. We consider an energy form

$$
{\cal E}(f,g) = \frac12\int_{\Omega} \nabla f\cdot \, \nabla \, g \, d\mu, \ \ \ f,g\in D,
$$
with $d\mu=\rho\, dx$, $\rho$ locally integrable, $\rho>0$ $dx$-a.e. and test functions D with compact support.
If $({\cal E}, D)$ is closable on $L^2(\Omega,\mu)$ and $\rho$ is regular enough, then
by {\it Fukushima's decomposition} the process
$((X_t)_{t\ge 0},(P_x)_{x\in \Omega})$ associated to ${\cal E}$ weakly solves
\begin{eqnarray*}
X_t & = & x + B_t+ \int_0^t \frac{\nabla \rho}{2\rho}(X_s)ds, \ \ \ P_x\mbox{-a.s for}\ x\in \Omega\setminus N,
\end{eqnarray*}
where $(B_t)_{t\ge 0}$ is a standard Brownian motion. Additional reflection terms may occur if $\rho$ is discontinuous or if we impose boundary conditions on $\partial \Omega$.
The set $N$ is in general an abstract capacity zero set and not explicitly known.
In this talk we investigate the stochastic regularity of $(X_t)_{t\ge 0}$, i.e. the question whether $N$ can be determined explicitly.
One of the difficulties is that $\frac{\nabla \rho}{2\rho}$ can be highly singular on the set $\{\rho =0\}$.
We consider different kind of weights and boundary conditions for which we can solve the problem. Our main tools are elliptic regularity results and local heat kernel estimates.
This talk is based on joint work with Jiyong Shin (SNU).

[Applied Mathematics]  조덕빈  Cho, Durkbin

• 동국대학교 수학과 교수
   Professor, Department of Mathematics, Dongguk University

Title: Overlapping Schwarz preconditioners for Isogeometric Analysis
Isogeometric Analysis (IGA) is a non-standard numerical method for partial differential equations (PDEs), which was introduced by T. J. R. Hughes in [1,2]. In the isogeometric framework, the ultimate goal is to adopt the geometry description from a Computer Aided Design (CAD) parametrization, and use it for the analysis, that is, within the PDE solver. Non-uniform rational B-splines (NURBS) are a standard in CAD community mainly because they are extremely convenient of the representation of free-form surfaces and there are very efficient algorithms to evaluate them, to refine and derefine them. In IGA, those same basis functions (that represent the CAD geometry) are also used as the basis for the discrete solution space of PDEs, thus following an isoparametric paradigm. IGA methodologies have been studied and applied in fields as diverse as fluid dynamics, structural mechanics and electromagnetics.

Domain decomposition methods are a major area of recent research in numerical analysis for PDEs. They provide robust, parallel and scalable preconditioned iterative methods for the large linear systems arising in discretizaton of the continuous problems.

In this talk, we give a brief review of B-splines, NURBS and Isogeometric Analysis and then propose overlapping additive Schwarz (OAS) methods for elliptic problems in Isogeometric Analysis [3]. We construct OAS preconditioners both in the parametric space and in the physical space and also prove that our proposed methods in multi-dimensions are scalable. Moreover, we present a set of numerical experiments, including the case with discontinuous coefficients, which is in complete accordance with the theoretical developments.

[1] T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg., 194(39-41):4135--4195, 2005.

[2] J.A. Cottrell, T.J.R. Hughes and Y. Bazilevs. Isogeometric Analysis: Toward Integration of CAD and FEA. John Wiley & Sons, 2009.

[3] L. Beirao da Veiga, D. Cho, L. Pavarino and S. Scacchi. Overlapping Schwarz Methods for Isogeometric Analysis. SIAM J. Numer. Anal., 50(3):1394--1416, 2012.

[Mathematical Education]  김부윤  Kim, Boo Yoon

• 부산대학교 수학교육과 교수
   Professor, Department of Mathematics Education, Pusan National University

Title: 수학교육의 여러 상황과 원인에 대하여(On the various situations and its roots in mathematics education)
수학교육 현장에 나타나는 여러 상황과 그러한 것이 발생하게 된 뿌리가 무엇인지에 대해 살펴본다. 특히 교육과정, 교과서 제도, 영재교육, 수학 축제, 대학입시, 대학수학교육,각급 학교 학생들의 변화 등에서 보이는 현재 상황으로는 참된 수학교육이 얼마나 어려운가를 던진다.

[Mathematics for Information Sciences]  권영수  Kwon, Young Soo

• 영남대학교 수학과 교수
   Professor, Department of Mathematics, Yeungnam University

Title: Symmetries of graph embeddings on surfaces
A map is a cellular embedding of a graph into a surface. Maps are expressed by several ways, for example, drawings, permutations, torsion free subgroups of Dyck's triangle groups, algebraic curves and so on. In this talk, we will consider relations among these expressions.  A map is called regular if its map automorphism group acts regularly on flag set which is the set of all incident vertex-edge-face triples. Classification of regular maps are mainly pursued by three different directions: for fixed graphs, fixed surfaces and for fixed groups. In this talk we give some methods to classify regular maps and survey some recent results related to classifications. If time admits,  some external symmetries of maps like exponents, self-duality and self-Petrie duality are also dealt with. 

[Cryptography]  서재홍  Seo, Jae Hong

• 명지대학교 수학과 교수
   Professor, Department of Mathematics, Myongji University
• 2012년도 상산 젊은수학자상 수상자
   The winner of 2012 Sangsan Prize for young Mathematicians

Title: (Im)possibility of Cryptographic Properties in Bilinear Product Groups
Since Boneh and Franklin used bilinear groups to design the first identity-based encryption scheme in 2001, bilinear groups has been used in various areas of cryptography. In 2005, Boneh, Goh, and Nissim found that product groups of bilinear groups have useful cryptographic properties, which are identified and named by Freeman later. In this talk, we introduce recent progress in this field; in particular, we focus on the (im)possibility of these cryptographic properties in prime-order setting.