2011 Ѽȸ ǥȸ

2011.4.30(), б
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<Plenary Lecture>

 

 

Shui-Nee Chow

 

Professor of Mathematics, Georgia Institute of Technology (USA)

 

Dynamical Systems and Nonlinear Studies

 

B.S. 1965, University of Singapore, Mathematics

Ph.D. 1970, University of Maryland, Mathematics

[Plenary Lecture]  ||  17:20~18:10, April 30(Sat), 2011


Fokker-Planck Equations for a Free Energy Functional or Markov Process on a graph

The classical Fokker-Planck equation is a linear parabolic equation which describes the time evolution of probability distribution of a stochastic process defined on an Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability
distributions and is a linear combination of a potential and an entropy. In recent years, it has been shown that Fokker-Planck equation is the gradient flow of the free energy functional defined on the Riemannian
manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance. In this paper, we consider analogous matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If $N\ge 2$ is the number of vertices of the graph, we show that the corresponding Fokker-Planck equation is a system of $N$ nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. However, in contrast to  stochastic processes defined on Euclidean spaces, situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different Fokker-Planck equations for the same process.  It is shown that there is a strong connection but also substantial discrepancies between the systems of ordinary differential equations and the classical Fokker-Planck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples will also be discussed. 

 

 

<Special Lecture>

 

  Se-Jung Oh

 

ѱ ̻ / President of National Research Foundation of Korea

 

[Special Lecture]  ||  16:30~17:20, April 30(Sat), 2011


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<Invited Lectures>

 

ݻȣ  Sangho Kum

 

ϴб / Professor of Chungbuk National University

 

[Invited Lecture]  Applied Mathematics  ||  10:10~10:50, April 30(Sat), 2011


Resolvent average on symmetric cones
Recently Bauschke et al. introduced a very interesting and new notion of proximal average in the context of convex analysis, and studied this subject systemically from various viewpoints. In addition, this new concept was applied to positive semidefinite matrices under the name of resolvent average, and basic properties of the resolvent average are successfully established by themselves from a totally different view and techniques of convex analysis rather than the classical matrix analysis. Inspired by their works and the well-known fact that the convex cone of positive definite matrices is a typical example of a symmetric cone
(self-dual homogeneous convex cone), we study the resolvent average on symmetric cones, and derive corresponding results in a different manner based on a purely Jordan algebraic technique. Some average for general convex functions is also introduced.  
 

 

Ǽ  Soonsik Kwon

 

KAIST / Professor of Mathematical Science, KAIST

2010 ڻ / The winner of 2010 Sangsan Prize for young mathematicians

 

[Invited Lecture]  Analysis  ||  15:20~16:00, April 30(Sat), 2011

 

Poincare-Dulac normal form reduction for unconditional well-posedness of the periodic cubic

We implement an infinite iteration scheme of Poincare-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrodinger equation. As a result, we prove its unconditional well-posedness in L^2. In the talk we explain how to adopt Poincare-Dulac normal form technique to PDE setting with multilinear estimates.

 

ڼȣ  Sung Ho Park

 

ѱܱб / Professor of Hankuk University of Foreign Studies

 

[Invited Lecture]  Geometry  ||  11:00~11:40, April 30(Sat), 2011

 

Symmetry of planar curves and capillary surface in a wedge in $\mathbb H^3$

We give several conditions for a planar curve to be a circle in $\mathbb R^2$ and $\mathbb H^2$, and apply the result to derive a necessary condition for capillary surfaces in a wedge in $\mthbb H^3$.

 

ɰ  Kyung-AH Shim

 

п ӿ / Senior researcher of National Institute for Mathematical Sciences of Korea

 

[Invited Lecture]  Cryptography  ||  15:20~16:00, April 30(Sat), 2011


Efficient ID-based signatures and multisignatures based on the q-SDH problem
Paterson and Schuldt proposed an ID-based signature scheme based on Waters' signature scheme, secure in the standard model under the Computational Diffie-Hellman assumption. They claimed that the
scheme allows secure ID-based aggregate signature (multisignature) scheme. We show that the ID-based aggregate (multisignature) scheme based on Paterson and Schuldt's signature scheme is insecure against
forgery attacks. We then propose a new ID-based signature scheme based on the q-SDH assumption which allows us to construct an efficient multisignature scheme secure in the standard model.  
 

 

  Chan Woo Yang

 

б / Professor of Korea University

 

[Invited Lecture]  Analysis  || 


Non-localization phenomena of oscillatory integral operators
In this talk we consider oscillatory integral operators $T_{\lambda}$ of the form $$T_{\lambda}f(x)=\int_{\mathbb{R}} e^{i \lambda S(x,y)} f(y) dy,$$ where $S$ is a polynomial with two real variables $x$ and $y$. We call these operators non-localized oscillatory integral operators. Usually oscillatory integral operators are operators of the form $$T_{\lambda}f(x)=\int_{\mathbb{R}} e^{i \lambda S(x,y)} \chi (x,y) f(y) dy,$$
where $\chi$ is a smooth cut-off function supported in a small neighborhood of the origin in $\mathbb{R}^2$. Because of the presence of the cut-off function $\chi$ we call these operators localized oscillatory integral operators. Almost full picture of decay rate of operator norm of localized oscillatory integral operators have been drawn in terms of Newton polygon of the phase function $S$ in numerous articles. However non-localized oscillatory integral operators have not been fully understood. The aim of this talk is to discuss interesting phenomena of mapping properties of non-localized oscillatory integral operators due to the absense of the cut-off function $\chi$ in terms of `non-localized' Newton polygon in adapted coordinate system.  
 

 

Ź  Young Tak Oh  


б / Professor of Sogang University

 

[Invited Lecture]  Algebra  ||  13:10~13:50, April 30(Sat), 2011


Witt vectors and Witt-Burnside rings 
The Witt-Burnside ring of a profinite group is a group-theoretical generalization of big Witt vectors. For instance, the classical $p$-typical Witt vectors of Teichmuller and Witt correspond to the Witt-Burnside ring of the profinite $p$-completion of the infinite cyclic group $C$ and the universal Witt vectors of Lang and Witt to that of the profinite completion of $C$.
In this talk, we explain recent results on the structure of Witt-Burnside rings. A special emphasis will be put on the classification and decomposition. Connection to representation theory will be also discussed.

 

Ӽ  Seonhee Lim

 

б / Professor of Seoul National University

 

[Invited Lecture]  Topology  ||  10:10~10:50, April 30(Sat), 2011

 

Dynamics of group action on non-positively curved spaces and its application

We introduce several dynamical properties of group action on non-positively curved spaces, and relate them with topological and geometrical properties of the spaces. We will also show some examples of its applications to problems in number theory.

 

  Jongmin Han

 

б / Professor of Mathematics, Kyung Hee University

 

[Invited Lecture]  Analysis  ||  13:10~13:50, April 30(Sat), 2011

 

Existence of topological solutions in the self-dual gauge field theories via super and subsolution method
In this talk, we discuss how to find topological solutions in the self-dual gauge field theories via the method of super and subsolutions. In particluar, we propose a new method to find an explicit subsolution. As an application, we provide a new proof of the existence of topological solutions in two Higgs model. We also prove the existence of topological solutions in the Maxwell gauged $O(3)$ sigma model and the Chern-Simons gauged $O(3)$ sigma model. 

 

Ȳ Sunwook Hwang

 

Ǵб / Professor of Soongsil University

 

[Invited Lecture]  Mathematical Education  ||  13:10~13:50, April 30(Sat), 2011


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Lucian Beznea

 

Institute of Mathematics of the Romanian Academy, Romania

 

[Invited Lecture]  Probability and Statistics  ||  11:00~11:40, April 30(Sat), 2011

 

On the quasi-regularity of non-sectorial Dirichlet forms by processes having the same polar sets

We obtain a criterion for the quasi-regularity of generalized (non-
sectorial) Dirichlet forms, which extends the result of P.J. Fitzsimmons on
the quasi-regularity of (sectorial) semi-Dirichlet forms.
Given the right (Markov) process associated to a semi-Dirichlet form, we present sufficient conditions for a
second right process to be a standard one, having the same state space.
The above mentioned quasi-regularity criterion is then an application.
The conditions are expressed in terms of the associated capacities, nests
of compacts, polar sets, and quasi-continuity.
A second application is on the quasi-regularity of the generalized
Dirichlet forms obtained by perturbing a semi-Dirichlet form with kernels. The talk is based on a
joint work with Gerald Trutnau.

 

Xuding Zhu

 

Professor of Zhejiang Normal University, China

 

[Invited Lecture]  Mathematics for Information Sciences  ||  14:00~14:40, April 30(Sat), 2011

 

Thue choice number of graphs

A sequence of even length is a repetition if the first half is identical to the second half. A sequence is said to contain a repetition if it has a subsequence which is a repetition. A classical result of Thue says that there is an infinite sequence on $3$ symbols which contains no repetition. This result motivated many deep research and challenging problems. One graph concept related to this result is Thue-colouring. A Thue-colouring of a graph $G$ is a mapping which assigns to each vertex of $G$ a colour (a symbol) in such a way that the colour sequence of any path of $G$ contains no repetition. The Thue-chromatic number of a graph is the minimum number of colours needed in a Thue-colouring of $G$. Thue's result is
equivalent to say that the infinite path has Thue-chromatic number $3$. It is also known that the Thue-chromatic number of any tree is at most $4$.
Thue-choice number of a graph $G$ is the list version of its Thue-chromatic number, which is the minimum integer $k$ such that if each vertex of $G$ is given $k$-permissible colours, then there is a Thue-colouring of $G$ using a permissible colour for each vertex. This talk will survey some research related to Thue Theorem and will show that Thue-choice number of paths is at most $4$ and Thue choice number of trees are unbounded.

 

 



 

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