It is an important conjecture that the Chow group of a smooth projective variety over a number field is a finitely generated Z-module, and no one knows how to solve it so far. On the other hand people considered two modified questions which are more accessible. One is whether the Chow group modulo n is finite or not, and the other is whether the torsion part is finite or not.

In this talk I survey recent progress about these two questions made by Shuji Saito, Kanetomo Sato and the speaker

The narrow escape problem consists of deriving the asymptotic expansion of the solution of a drift-diffusion equation with the Dirichlet boundary condition on a small absorbing part of the boundary and the Neumann boundary condition on the remaining reflecting boundaries. Using layer potential techniques, we rigorously find high-order asymptotic expansions of such solutions. The asymptotic formula explicitly exhibit the nonlinear interaction of many small absorbing targets. 

I will talk about problems on a Siegel type domain in the complex Euclidean space. Here the Siegel type domain means the Siegel domain of first and second type in the sense of Pyatetskii-Shapiro or a domain defined by a weighted homogeneous polynomial. It has been known that most of model objects in Complex Geometry of negatively curved Kaehler manifolds or Complex Analysis of bounded domains are biholomorphic to Siegel type domain. Although Siegel type domains are unbounded in C^n, they have many affine automorphisms, so the Siegel type realization of models has been employed in wide area of Several Complex Variables. In this talk I will discuss a geometric problem on the realization of a Siegel type domain as a bounded domain.

 It is clear that the physically meaningful quantity is not the potential itself, but its difference or gradient. However, the theory is based on the potential itself. In this talk the speaker will introduce new concept of relative potential.

Quasi-random graphs can be informally described as graphs whose edge distribution closely resembles that of a random graph. They have been a subject of intensive study during the last two decades and have seen numerous applications both in Combinatorics and Computer Science.

Starting with the work of Thomason and Chung, Graham, and Wilson, there have been many works which established the equivalence of various properties of graphs to quasi-randomness. In this talk, I will give a survey on this topic, and provide a new condition which guarantees quasi-randomness. This result answers an open question raised independently by Janson, and Shapira and Yuster.

Joint work with Hao Huang (UCLA).

During last few years, there has been extensive study on 'Optimal mass transportation' problem. In this talk, we will describe some theories on this area and show how these can be applied to (evolutionary) Partial Differential Equations.

In this talk we introduce the theory of graph skein modules associted with the Yamada polynomial. We compute the graph algebra in some simple cases. Then, we apply graph skein techniques to establish necessary conditions for a spatial graph to have a symmetry of order p, where p is a prime.

Self-dual codes have become one of the most active research areas in coding theory due to their rich mathematical theories. In this talk, we start with an introduction to coding theory. Then we describe some recent results on the constructions of self-dual codes over rings, and applications to lattices and network coding theory. We conclude the talk with some open problems.

The Riordan group is an easy yet powerful tool for looking at a large number of results in combinatorial enumeration. At the first level it provides quick proofs for many binomial identities as well as a systematic way to invert them. We will see how they arise naturally when looking at the uplift principle as applied to classes of ordered trees. We will also discuss some recent results including the Double Riordan group, summer – winter trees, spoiled child trees, and will mention a few open problems as well. The main tools involved are generating functions, matrix multiplication, and elementary group theory.

The aim of this talk is to introduce various free boundary problems
including a shock problem and applications. Even though the shock
problem is originally raised in fluid dynamics, its mathematical study
for the last several decades has brought more rigorous understanding
of the subject. Moreover, its application reaches to not only fluid
dynamics, engineering but also other area of mathematics as well. I
will discuss about applications of free boundary problem with
examples. If time is allowed, I will also show interesting free
boundary problems raised from fluid dynamics, and explain general
mathematical theory which can be obtained from these problems.

The chromatic polynomial of a graph counts the number of proper colorings of the graph. We give an affirmative answer to the conjecture of Read (1968) and Welsh (1976) that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. We define a sequence of numerical invariants of projective hypersurfaces analogous to the Milnor number of local analytic hypersurfaces. Then we show log-concavity of the sequence by answering a question of Trung and Verma on mixed multiplicities of ideals. The conjecture on the chromatic polynomial follows as a special case.

http://mathsci.kaist.ac.kr/~manifold/Arithmetics.html

In this talk, we first introduce the weakly over-penalized symmetric interior penalty (WOPSIP) method for second order elliptic problems, which belongs to the family of discontinuous Galerkin methods. Similar to the classical nonconforming P1 finite element method, this method satisfies the same types of error estimates as the standard conforming finite element method in both the energy norm and the L2 norm. Moreover, the WOPSIP method is more flexible than the classical nonconforming P1 finite element method in the sense that it can be implemented on meshes with hanging nodes.

Secondly, we discuss two-level additive Schwarz preconditioners for the WOPSIP method. The key ingredient of the two-level additive Schwarz preconditioner is the construction of the subdomain solvers and the coarse solver. In our approach, we consider different choices of coarse problems and intergrid transfer operators. It is shown that the condition number estimates previously obtained for classical finite element methods also hold for the WOPSIP method. In addition, we present numerical results that illustrate the parallel performance of these preconditioners.

In this talk, I will introduce a transonic shock problem for the Euler system of inviscid compressible flow, and explain how a transonic shock problem is formulated as a free boundary problem containing nonlinear mixed type PDEs.  The first part of the talk will be devoted to an introduction to various transonic shock problems including the shock reflection and the de Laval nozzle flow. In the second part of the talk, I will present the recent results about transonic shocks in multidimensional divergent nozzles. This is a joint work with Mikhail Feldman.