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.

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

We are considering a gas or "plasma" of electrons in a plane, in the presence of an external field which is strong enough near infinity that the particles be confined to a finite portion of it. If we let the number of particles increase indefinitely, the gas will condensate on a certain compact subset of the plane, known as the "droplet". The shape of this droplet depends on the external field. The problem to determine the details of it is the Laplacian growth problem from fluid mechanics. This is what gives the classical equilibrium distribution of the plasma. Looking at the gas in further detail, the first thing to observe is that the repulsion between the electrons will cause a "very uniform distribution" in the vicinity of the droplet. One of our theorems assert that the fluctuations about the equilibrium converges to a Gaussian field on the droplet with free boundary conditions. If there is time, I will also mention a new kind of field approximations which we can use to justify much of the physical formalism of conformal field theory. Joint work with Håkan Hedenmalm, Nikolai Makarov, and Nam-Guy Kang, in different constellations.

During the last decade, inverse combinatorial optimization problems have found an increased interest in the optimization community. Whereas an optimization problem asks for a feasible solution with minimum or maximum objective function value, inverse optimization problems are defined with a feasible solution, and aim to perturb as little as possible the parameters (costs, profits, etc.) of the problem so that the given solution becomes optimum in the new instance. In this talk, we introduce some generalized inverse combinatorial problems, and investigate inverse chromatic number problems in permutation graphs and interval graphs.