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 <Plenary Lecture>

 

Don Zagier  [Homepage]

 

Director, Max Planck Institute for Mathematics (Germany)

Professor, Collège de France, Paris (France)

Don Zagier received bachelor's degrees in mathematics and physics from M.I.T. at the age of 16 and a doctoral degree in mathematics from Oxford at the age of 20 under the supervision of Friedrich Hirzebruch at Bonn, with whom he later collaborated in work on Hilbert modular surfaces. He has held professorships in the Univerisities of Maryland, Kyushu, and Utrecht and is currently a Scientific Member and Director of the Max Planck Institute for Mathematics in Bonn and a Professor at the College de France, Paris. His main area of research is the theory of modular forms and their applications to other fields of mathematics. In particular, in joint work with Martin Eichler he developed the theory of Jacobi forms, which have proved to be a useful tool in several domains of number theory and theoretical physics.

[Plenary Lecture] April 28(Sat), 2012

 

Quantum invariants of knots and number theory

 

Ideas coming originally from quantum field theory, and in particular from the theory of quantum groups, have led to the definition of many new invariants of knots and 3-manifolds.  It turns out that these invariants have unexpectedly deep number-theoretical properties related in particular to algebraic $K$-theory and, even more surprisingly, to the theory of modular forms, though most of these relationships are still only conjectural.  In the lecture I will describe some of the examples that have led to these conjectures and some of the ideas behind them. 

 

 

<Invited Lectures>

 

Henry Kim

 

Professor, University of Toronto

 

[Invited Lecture - Recent Trends in Number Theory] April 28(Sat), 2012

 

Application of the strong Artin conjecture to the class number problem

 

As an application of the strong Artin conjecture, we exhibit unconditionally a family of number fields with extreme class numbers whose normal closures have S_5, S_4, A_4, and dihedral groups D_n, n=3,4,5, and cyclic groups C_n, n=4,5,6, as their Galois groups. This is a joint work with P.J. Cho.

 

ÀÌÁ¤¿¬  Lee, Jungyun

 

°íµî°úÇпø ¼öÇкΠ¿¬±¸¿ø  Research Fellow, Korea Institute for Advanced Study (KIAS)

2011³âµµ »ó»ê ÀþÀº¼öÇÐÀÚ»ó ¼ö»óÀÚ The winner of 2011 Sangsan Prize for young Mathematicians

 

[Invited Lecture - Recent Trends in Number Theory] April 28(Sat), 2012

 

Special values of Hecke L-function of real quadratic fields and class number problem of real quadratic fields

 

It is beyond our knowledge if there exist infinitely many real quadratic fields with class number one. This is mainly due to regulators that are not under control of discriminants.So most of the achievements in class number problem of real quadratic fields are in the restricted form where the regulator is controlled in a way. A best known such family is namely the Richaud-Degert type. In this family, as in the imaginary quadratic fields case, it has been known that there are only finitely many real quadratic fields with given class number. Moreover under the assumption of the generalized Riemann hypothesis, for a given class number, an explicit upper bound of discriminant of Richaud-Degert type can be obtained. Recent progress made by Biro, Byeon and myself suggests a way to obtain explicit upper bound of discriminant for some families of Richaud-Degert type with class number one, not assuming the generalized Riemann hypothesis.
For this we used a property that the values of Hecke-L-function at $s=0$ for some families of Richaud-Degert type are expressed as quasi-polynomial of degree 1.

Later I discovered that such behavior is closely related to the expression of continued fraction expansion. If we consider the following family of real quadratic fields $K_n=\Bbb{Q}(\sqrt{f(n)})$ for $f(x)\in \Bbb{Z}[x]$ and an ideal $\bf{b}_n$ of $K_n$ such that $$\bf{b}_n^{1}\sim[1,\omega(n)]$$ and
$$\omega(n)-1=[a_0(n),a_1(n),\cdots,a_{s-1}(n)]$$
with $a_i(x)\in\Bbb{Z}[X]$. We find that if $a_i(x)$ are all polynomial functions with degree $\leq d$ then the special values $L_{K_n}(0,\chi_n,\bf{b}_n)$ are expressed as quasi polynomial in $n$ of degree $\leq d$. Moreover the family $K_n$ has the regulator in control because of fixed length of continued fraction.This observation suggests families satisfying such a condition generalizes the R-D type.

This is a joint work with Byungheup Jun.

 

¹è¸íÁø  Bae, Myoung Jean

 

Æ÷Ç×°ø°ú´ëÇб³ ¼öÇаú Á¶±³¼ö  Assistant Professor, POSTECH

2011³âµµ »ó»ê ÀþÀº¼öÇÐÀÚ»ó ¼ö»óÀÚ  The The winner of 2011 Sangsan Prize for young Mathematicians

 

[Invited Lecture - Analysis] April 28(Sat), 2012

 

Nonlinear degenerate elliptic equation and its application

 

Motion of inviscid compressible flow is governed by Euler system. In self-similar flow of Euler system, a sonic arc appears when a shock occurs. Mathematically, a sonic arc is a boundary on which ellipticity or hyperbolicity of a nonlinear equation degenerates. This talk will present a regularity result of solutions to a class of nonlinear degenerate elliptic equations, and explain how this result is applied to establish optimal regularity of weak solutions to Euler system for self-similar flow on sonic arcs.

 

È«ÀçÇö Hong, Jaehyun

 

¼­¿ï´ëÇб³ ¼ö¸®°úÇкΠÁ¶±³¼ö  Assistant Professor, Seoul National University

 

[Invited Lecture - Geometry] April 28(Sat), 2012

 

Rigidities, geometric structures, and Lie algebra cohomologies

 

I will talk about how to solve a rigidity problem by transforming it into an equivalence problem of geometric structures and by computing the Lie algebra cohomology related to it.  

 

°í±âÇü  Ko, Ki Hyoung

 

Ä«À̽ºÆ® ¼ö¸®°úÇаú ±³¼ö  Professor, Korea Advanced Institute of Science and Technology (KAIST)

2011³âµµ ´ëÇѼöÇÐȸ Çмú»ó ¼ö»óÀÚ  The winner of KMS Academic Achievement Award

 

[Invited Lecture - Topology] April 28(Sat), 2012

 

A cycle of frustration and rewarding moments

 

I began my career as a mathematician in 1984 after PhD at Brandeis. It is a great pleasure to retrospect what I have done since then as a researcher in theory of knots and braids---especially those that are memorable to me. I thank KMS for providing me this opportunity.

 

°­³²±Ô  Kang, Nam-Gyu

 

¼­¿ï´ëÇб³ ¼ö¸®°úÇкΠÁ¶±³¼ö  Assistant Professor, Seoul National University

 

[Invited Lecture - Probability and Statistics] April 28(Sat), 2012

 

Distribution of eigenvalues of random normal matrices near the edge of the spectrum

 

Microscopic properties of eigenvalues of random normal matrices change drastically in a narrow belt around the edge of the spectrum. I present an elementary method to prove Borodin and Sinclair's theorem on the scaling limit of correlation kernels for the soft-edge Ginibre ensemble. This method gives new result for the hard-edge Ginibre ensemble. After a discussion of the general properties of this scaling limit, I state a universality conjecture and provide arguments to support it. This is a joint work with Y. Ameur and N. Makarov.  

 

Ç¥ÀçÈ«  Pyo, Jae-Hong

 

°­¿ø´ëÇб³ ¼öÇаú Á¶±³¼ö  Assistant Professor, Kangwon National University

 

[Invited Lecture - Applied Mathematics] April 28(Sat), 2012

 

Error estimates for the second order semi-discrete stabilized Gauge-Uzawa method for the Navier-Stokes equations

 

The Gauge-Uzawa method [GUM], which is a projection type algorithm to solve the time depend Navier-Stokes equations, has been constructed in [2] and enhanced in [3, 5] to apply to more complicated problems. Even though GUM possesses many advantages theoretically and numerically, the studies on GUM have been limited on the first order backward Euler scheme except normal mode error estimate in [4]. The goal of this paper is to research the 2nd order GUM. Because the classical 2nd order GUM which is studied in [4] needs rather strong stability condition, we modify GUM to be unconditionally stable method using BDF2 time marching. The stabilized GUM is equivalent to the rotational form of pressure correction method and the errors are already estimated in [1] for the Stokes equations. In this paper, we will evaluate errors of the stabilized GUM for the Navier-Stokes equations. We also prove that the stabilized GUM is an unconditionally stable method for the Naiver-Stokes equations. So we conclude that the rotational form of pressure correction method in [1] is also unconditionally stable scheme and that the accuracy results in [1] are valid for the Navier-Stokes equations.

[1] J. L. Guermond and J. Shen, On the error estimates of rotational pressure-correction
projection methods
, Math. Comp. 73 (2004), 1719--1737.

[2] R. H. Nochetto and J.-H. Pyo, A finite element Gauge-Uzawa method. Part I : the Navier-Stokes equations, SIAM J. Numer. Anal. 43 (2005), 1043--1068.

[3] R. H. Nochetto and J.-H. Pyo, A finite element Gauge-Uzawa method. Part II : Boussinesq Equations, Math. Models Methods Appl. Sci. 16 (2006), 1599--1626.

[4] J.-H. Pyo and J. Shen, Normal Mode Analysis of Second-order Projection Methods for Incompressible Flows, Discrete Contin. Dyn. Syst. Ser. B 5 (2005), 817--840.

[5] J.-H. Pyo and J. Shen, Gauge Uzawa methods for incompressible flows with Variable Density, J. Comput. Phys. 211 (2007), 181--197.

 

Á¶¼öÁø  Cho, Soojin

 

¾ÆÁÖ´ëÇб³ ÀÚ¿¬°úÇкΠ¼öÇÐÀü°ø ºÎ±³¼ö  Associate Professor, Ajou University

 

[Invited Lecture - Mathematics for Information Sciences] April 28(Sat), 2012

 

Skew Schur P-functions 

 

There are two combinatorial models for Schur P-functions; marked shifted tableaux and semistandard decomposition tableaux. Skew marked shifted tableaux can be defined in a natural way to define skew Schur P-functions, yet a natural definition of skew semistandard decomposition tableaux is not known.
A model for the skew semistandard decomposition tableaux will be given while the falsity of the conjecture by L. Serrano is explained. 
 

 

Robert Beezer

 

Professor, University of Puget Sound (USA)

 

[Invited Lecture - Mathematical Education] April 28(Sat), 2012

 

Some overview of Sage and its use in teaching linear algebra, abstract algebra and graph theory with Open Textbooks

 

Sage is an open source software system for advanced mathematics, and is a natural choice for use in teaching.
I will describe its use in teaching linear algebra, abstract algebra and graph theory, along with open textbooks.

 

ÀÌÁÖ¿µ  Lee, Jooyoung

 

¼¼Á¾´ëÇб³ ÀÀ¿ë¼öÇаú Á¶±³¼ö  Assistant Professor, Sejong University

 

[Invited Lecture - Cryptography] April 28(Sat), 2012

 

Security of the Even-Mansour scheme and its generalization

 

In this talk, we are going to survey the recent results on the Even-Mansour scheme and its generalization. We give an alternative security proof for the 2-round key alternating construction, and show that the key size can be reduced from 3 keys to 2 keys.  

 

 




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