2012 Ѽȸ ȸ ǥȸ

2012.10.5()~6(), Ǽ
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<Open KIAS (Open KIAS Plenary Lecture)>

15:20~16:10, October 5(fri), 2012

 

̿볲  Lee, Yongnam  [Homepage]

 

ī̽Ʈ а

Professor, Department of Mathematical Sciences,

Korea Advanced Institute of Science and Technology (KAIST)

 

о: Algebraic geometry

 

1987. 2 б л B.S. Seoul National University

1989. 2 б M.S. Seoul National University

1997. 6 ̱ Ÿб ڻ Ph.D University of Utah (USA)

$\mathbb Q$-Gorenstein deformation theory and its applications

In this talk, I will review singularities of class T and $\mathbb Q$-Gorenstein deformation theory. Construction of simply connected surfaces of general type with $p_g=q=0$ via $\mathbb Q$-Gorenstein smoothings over the field of any characteristic will be also reviewed. Most works in this talk have been carried out by the joint research with Jongil Park and with Noboru Nakayama.

The notion of singularity of class T, which is defined as a quotient surface singularity admitting a $\mathbb Q$-Gorenstein smoothing, was introduced by Koll\'ar and Shepherd-Barron [2]. They also gave an explicit description of the singularity of class T. Previous study for singularities for class T and $\mathbb Q$-Gorenstein deformation theory use the property that an index one cover is \'etale on the Gorenstein locus, which does not hold in characteristic $p > 0$ when the Gorenstein index is divisible by $p$. By developing the theory of $\mathbb Q$-Gorenstein deformation functor, we can generalize their results to surfaces in positive characteristics. In [5] we prove the existence of versal $\mathbb Q$-Gorenstein $\Lambda$-deformation for some typical $\mathbb Q$-Gorenstein $k$-schemes for a fixed complete Noetherian ring $\Lambda$ with an algebraically closed residue field $k$.

After the paper [3], which constructs a simply connected Campedelli surface (a minimal projective surface of general type with $p_g=q=0$, $K^2=2$) by using a $\mathbb Q$-Gorenstein smoothing and Milnor fiber of a smoothing (or rational blow-down surgery), several interesting examples of surfaces of general type with $p_g=q=0$ were constructed via $\mathbb Q$-Gorenstein smoothings. Now, these $\mathbb Q$-Gorenstein smoothing methods are extended to some other type of surfaces and to surfaces in positive characteristics. To find a new family of simply connected surfaces of general type with $p_g=q=0$ is one of the fundamental problems in the study of algebraic surfaces. Surfaces with $p_g =q= 0$ are interesting in view of Castelnuovos criterion: An irrational surface with $q = 0$ must have $P_2\ge 1$. This class of surfaces has been studied extensively by algebraic geometers and topologists for a long time. In particular, simply connected surfaces of general type with$p_g=0$ are little known. Before the paper [3] the only previously known simply connected, minimal, complex surface of general type with $p_g=0$ was Barlow surface [1]. Barlow surface has $K^2 = 1$. When a surface is defined over a field of positive characteristic, the existence of algebraically simply connected minimal surface of general type with $p_g = 0$ is known only for some special characteristics. The paper [4] constructs such a surface of general type defined over an algebraically closed field of any characteristic applying the construction given in [3]. Construction in [3] is as follows:

First, we consider a special pencil of cubics in $\mathbb P^2$ and blow up many times to get a projective surface $M$, which contains a disjoint union of five linear chains of smooth rational curves representing the resolution graphs of singularities of class $T$. Then, we contract these linear chains of rational curves from the surface $M$ to produce a projective surface $X$ with five singularities of class T and with $K^2_X = 2$. We can prove the existence of a global $\mathbb Q$-Gorenstein smoothing of the singular surface $X$, in which a general fiber $X_t$ of the $\mathbb Q$-Gorenstein smoothing is a simply connected minimal surface of general type with $p_g = 0$ and $K^2 = 2$. This method of construction works to other types of rational elliptic surfaces, which are used to construct a simply connected minimal surface of general type with $p_g = 0$ and with $K^2 = 1, 3$, or $4$. The paper [4] shows that the construction of singular surfaces also works in positive characteristic, but several key parts in the proof to show the existence of a global $\mathbb Q$-Gorenstein smoothing should be modified. We construct a deformation of a normal projective surface $X$ with toric singularities of class T assuming some extra conditions. As a consequence, we have a so-called $\mathbb Q$-Gorenstein smoothing not only over the base field $k$ but also over a complete discrete valuation ring with the residue field $k$. By the smoothing over the discrete valuation ring and the Grothendieck specialization theorem, the algebraic simply connectedness of the smooth fiber is reduced to that of a smooth fiber of a $\mathbb Q$-Gorenstein smoothing of a reduction of $X$ of our construction to the complex number field.

 

[1] R. Barlow, A simply connected surface of general type with $p_g =0$, Invent. Math. 79 (1984), 293--301.

[2] J. Koll\'ar and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299--338.

[3] Y. Lee and J. Park, A simply connected surface of general type with $p_g=0$ and $K^2=2$, Invent. Math. 170 (2007), 483--505.

[4] Y. Lee and N. Nakayama, Simply connected surfaces of general type in positive characteristic via deformation theory, to appear in PLMS.

[5] Y. Lee and N. Nakayama, $\mathbb Q$-Gorenstein deformation theory over a complete Noetherian local ring, In preparation.

 

 

<ѱл (Special Lecture by the Korean Science Award winner)>

16:20~17:10, October 5(fri), 2012

 

 

  Park, Jongil

 

б к

Professor, Department of Mathematical Sciences,
Seoul National University

 

о: -4 پü

 

1986. 2 б л B.S. Seoul National University

1988. 2 б M.S. Seoul National University

1996. 8 ̱ ̽ðָб ڻ Ph.D Michigan State University (USA)

Simply connected symplectic 4-manifolds with $b_{2}^{+}=1$ and $c_{1}^{2}=2$ - Before and After 2004

One of the fundamental problems in the topology of 4-manifolds is to classify all smooth (symplectic, complex) 4-manifolds homeomorphic to the same underlying topological 4-manifold. Even though gauge theory has been very successful in addressing this question, the complete answer is still far from reach. In particular, for smooth structures on simply connected closed smooth $4$-manifolds, most known results have been obtained for the case of 4-manifolds with either $b_2^+ > 1$ and odd or $b_2^+ = 1$ and $c_1^2 \leq 0$.  In the case of $b_2^+ = 1$ and $c_1^2 > 0$, the only known result is the theorem of D. Kotschick that the Barlow surface ([1]) is not diffeomorphic to ${\mathbf CP}^2#8{\overline{{\mathbf CP}}^2}$ ([3]).

Since I discovered a new simply connected symplectic $4$-manifold with $b_2^+ =1$ and $c_1^2=2$ ([5]) in 2004 by using a rational blow-down surgery, many new simply connected $4$-manifolds with small Euler characteristic have been constructed ([2], [4], [6], [7], [8], [9]) and now it is one of the most active research areas in $4$-manifolds to find a new family of smooth (symplectic, complex) $4$-manifolds with $b_2^+ =1$ and $c_1^2 > 0$.

The aim of this talk is to review briefly how to construct such a symplectic $4$-manifold with $b_2^+ =1$ and $c_1^2=2$. And then I'll explain how this construction affects the study of simply connected $4$-manifolds with $b_2^+=1$ (equivalently, $p_g=0$ in complex category) in three levels - smooth category, symplectic category and complex category.

 

[1] R. Barlow, A simply connected surface of general type with $p_g =0$, Invent. Math. 79 (1984), 293--301

[2] R. Fintushel and R. Stern, Double node neighborhoods and families of simply connected 4-manifolds with $b^+=1$, J. Amer. Math. Soc. 19 (2006), 171--180

[3] D. Kotschick, On manifolds homeomorphic to ${\mathbf CP}^2#8{\overline{{\mathbf CP}}^2}$, Invent. Math. 95 (1989), 591--600

[4] Y. Lee and J. Park, A surface of general type with $p_g=0$ and $K^2=2$, Invent. Math. 170 (2007), 483--505

[5] J. Park, Simply connected symplectic 4-manifolds with $b_2^+=1$ and $c_1^2=2$, Invent. Math. 159 (2005), 657--667.

[6] H. Park, J. Park, and D. Shin, A simply connected surface of general type with $p_g=0$ and $K^2=3, 4$, Geom. Topol. 13 (2009), 743--767; 1483--1494.

[7] J. Park, A. Stipsicz, and Z. Szabò, Exotic smooth structures on ${\mathbf CP}^2#5{\overline{{\mathbf CP}}^2}$, Math. Res. Letters 12 (2005), 701--712

[8] A. Stipsicz and Z. Szabò, An exotic smooth structure on ${\mathbf CP}^2#6{\overline{{\mathbf CP}}^2}$, Geom. Topol. 9 (2005), 813--832

[9] A. Stipsicz, Z. Szabò, and J. Wahl, Rational blowdowns and smoothings of surface singularities, J. Topology 1 (2008), 477—517

 

 

<û(Invited Lectures)>

[Algebra] 13:40~14:20, October 5(fri), 2012

 Kang, Soon-Yi

 

б а α

Associate Professor, Department of Mathematics, Kangwon National University

 

Mock modular forms of small weights

One can construct a harmonic weak Maass form using weak Maass-Poincar\'{e} series, but for weights between 0 and 2, care needs to be taken as the Poincar\'{e} series does not converge. Many interesting examples of harmonic weak Maass forms or mock modular forms are found in these weights though, such as Ramanujan's mock theta functions, traces of singular moduli, Niebur-Poincar\'{e} series, generating functions of partitions or compositions. In this talk, we present these mock modular forms and various properties of them.

 

 

[Analysis-1] 14:30~15:10, October 5(fri), 2012

 Kim, Namkwon

 

б а

Assistant Professor, Department of Mathematics, Chosun University

  

Equivariant degree formula and gauge theory

We consider some gauge theory here and present existence results for PDEs in those gauge theory. The relevant PDE problem here is basically nonlinear and nonmonotone(That is, certain crossing happens). The equivariant degree formula gives nice framework for these problems and will be presented.

 

 

[Analysis-2] 13:40~14:20, October 5(fri), 2012

߿  Heo, Ya Ryong

 

б а

Assistant Professor, Department of Mathematics, Korea University

 

2012 Ѽȸ

The winner of 2012 Excellent Research Paper Award

 

Radial Fourier multipliers in high dimensions

This is a joint work with F. Nazarov and A. Seeger. A simple characterization of convolution operators bounded on $L^p$ is known only in two cases: $p=1$ and $p=2$. We will deal with convolution operators with radial kernels acting on functions defined in $R^d$. And we will try to give a simple characterizations of radial functions that are multipliers in $L^p$, $p \neq 2$.

 

 

[Geometry] 13:40~14:20, October 5(fri), 2012

  Choe, Jaigyoung

 

п к

Professor, Department of Mathematics, Korea Institute for Advanced Study (KIAS)

 

Higher dimensional versions of the Enneper surface, catenoid and helicoids

The Enneper surface, catenoid and helicoid are the simplest complete minimal surfaces in $\mathbb R^3$. They can be constructed by applying the Weierstrass representation formula. But there is no such formula for higher dimensional minimal surfaces. However, using some geometric properties, we construct the higher dimensional generalizations of these surfaces in $\mathbb R^n$.

 

 

[Topology-1] 10:40~11:20, October 5(fri), 2012

ö  Cho, Cheol-Hyun

 

б к α

Associate Professor, Department of Mathematics, Seoul National University

 

2012 Ѽȸ

The winner of 2012 Excellent Research Paper Award

 

Finite group action, Floer theory and orbifolds

This will be an introductory talk regarding Lagrangian Floer theory on orbifolds, its homological mirror symmetry, and related phenomenons. In particular, the role of group actions, and orbifold holomorphic curve will be explained, as well as open crepant resolution conjecture.

 

 

[Topology-2] 11:30~12:10, October 5(fri), 2012

Kenichi Ohshika

 

Osaka University (Japan)

 

Deformation spaces of Kleinian groups and beyond

The main object in Kleinian group theory is now studying topological structure of deformation spaces of Kleinian groups.

Each deformation space is embedded in a character variety, and it is also an important problem to determine how it is embedded and what kind of representations are nearby.

In this talk, I shall discuss recent progresses in this field including my own work on boundaries of deformation spaces.

 

 

[Probability and Statistics] 12:00~12:40, October 6(sat), 2012 

Zoran Vondraček

 

University of Zagreb (Republic of Croatia)

 

Potential theory of subordinate Brownian motions with Gaussian components

In this paper we study a subordinate Brownian motion with a Gaussian component

and a rather general discontinuous part. The assumption on the subordinator

is that its Laplace exponent is a complete Bernstein function with a L\'evy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in $C^{1,1}$ open sets.

As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded $C^{1,1}$ open set $D$ and identify the Martin boundary of $D$ with respect to the subordinate Brownian motion with the Euclidean boundary.

 

 

[Applied Mathematics] 13:40~14:20, October 5(fri), 2012

  Hyon, YunKyong

 

п

Research Scientist, National Institute for Mathematical Sciences of Korea (NIMS)

 

Mathematical modeling of ionic fluids in ion channel: energetic variational approach

We discuss a mathematical model for the transport of ions through ion channels. The resulting system of partial differential equations is derived in the frame of the energetic variational approaches, taking into account the coupling between electrostatics, diffusion and protein (ion channel) structure. We also incorporate the geometric constraints of the ion channel through a potential energy controlling

the local maximum volume inside the ion channel. In the mathematical modeling,

a diffusive interface (labeling) description is also employed to describe the geometric configuration of the channels. The energy functional consists of the entropic free energy for diffusion of the ions, the electrostatic potential energy, the repulsive potential energy for the excluded volume effect of the ion particles and the potential energy for the geometric constraints of the ion channel. For the biological application of such a system, we consider channel recordings of voltage clamp to measure the current flowing through the ion channel. The results of one-dimensional numerical simulations are presented to demonstrate some signature effects of the channel.

 

 

[Mathematical Education] 11:30~12:10, October 5(fri), 2012

ȫ  Kim, Hong-Jong

 

б к

Professor, Department of Mathematics, Seoul National University

 

Harmonic mean

In everyday life, we see, hear, and feel harmony. I will explain some of them.

 

 

[Mathematics for Information Sciences] 10:40~11:20, October 5(fri), 2012

  Oum, Sang-il

 

ī̽Ʈ а α

Associate Professor, Department of Mathematical Sciences,

Korea Advanced Institute of Science and Technology (KAIST)

 

Even cycle decomposition of graphs with no odd K_4 Minor

An even cycle decomposition of a graph $G$ is a partition of $E(G)$ into cycles of even length. Evidently, every Eulerian bipartite graph has an even cycle decomposition. Seymour [circuits in planar graphs.

J. Combin. Theory Ser. B 31 (1981), no. 3, 327--338] proved that every $2$-connected loopless Eulerian planar graph with an even number of edges also admits an even cycle decomposition. Later, Zhang [On even circuit decompositions of {E}ulerian graphs.

J. Graph Theory 18 (1994), no. 1, 51--57] generalized this to graphs with no $K_5$-minor.

In this paper we propose a conjecture involving signed graphs which contains all of these results. Our main result is a weakened form of this conjecture. Namely, we prove that every $2$-connected loopless Eulerian odd-$K_4$-minor free signed graph with an even number of odd edges has an even cycle decomposition.

 

 

[Cryptography] 10:40~11:20, October 5(fri), 2012

ڿȣ  Park, Young-Ho

 

̹б ȣа

Professor, Department of Information Security, Sejong Cyber University

 

Side channel attacks and their applications

In cryptography, a side channel attack is any attack based on side channel information gained from the physical implementation of a cryptosystem. It takes advantage of implementation-specific characteristics to recover the secret parameters involved in the computation. It is therefore much less general but often much more powerful than mathematical cryptanalysis. In this talk, we are going to survey side channel attacks and their applications.

 

 


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