Lascoux-Leclerc-Thibon conjectured the simple modules over Hecke akgebras
are controled by the fundamental representation of affine quantum groups.
This is proved by Ariki in a more generalized form.
Recently, Kovanov-Lauda and Rouquier introduced a new algebra
which is generalization of Hecke algebras,
and they conjectured that this algebra categorifies highest modules
of quantum groups. This conjecture is solved by myself and
Seok-Jin Kang (arXiv:1102.4677). In this talk, I will explain them.

There are three ways to define an orbifold: orbifolds
charts, groupoids, stacks. I will explain these notions in smooth
category by working out a few basic examples. It will be one of the
goals of the talk to give some understanding of the notion of stacks,
which is an important language to study moduli problems and also an
orbifold as a presentation-free intrinsic object. If time allows, I
will get to the definitions of group actions on orbifolds and the
(equivariant) cohomology over integer coefficients.

critical  exponent for the Sobolev embedding  of H 1 (Ω) into Lq (Ω).  We discuss some

existence  results  for the problem  above.

A set A of integers is a Sidon set if all the sums a₁+a₂, with a₁≤a₂and a₁, a₂∈A, are distinct. In the 1940s, Chowla, Erdős and Turán showed that the maximum possible size of a Sidon set contained in [n]={0,1,…,n-1} is √n (1+o(1)). We study Sidon sets contained in sparse random sets of integers, replacing the ‘dense environment’ [n] by a sparse, random subset R of [n].

Let R=[n]_m be a uniformly chosen, random m-element subset of [n]. Let F([n]_m)=max {|S| : S⊆[n]_m Sidon}. An abridged version of our results states as follows. Fix a constant 0≤a≤1 and suppose m=m(n)=(1+o(1))n^a. Then there is a constant b=b(a) for which F([n]_m)=n^b+o(1) almost surely. The function b=b(a) is a continuous, piecewise linear function of a, not differentiable at two points: a=1/3 and a=2/3; between those two points, the function b=b(a) is constant. This is joint work with Yoshiharu Kohayakawa and Vojtech Rödl.

 In analytic number theory it is an interesting problem to find sharp  asymptotic bound for partial sum of Moebius function, due to its  intimate connection with Riemann zeta function. After we sketch Nathan  Ng's heuristic approach (2004) to this problem, we will consider its  function field analogue, in the context of a function field of algebraic  curves over finite fields. 

Petersen proved that every cubic graph without cut-edges has a perfect matching, but some graphs with cut-edges have no perfect matching. The smallest cubic graph with no perfect matching belongs to a general family applicable to many problems on connected d-regular graphs with n vertices. These include the smallest matching number for such graphs and a relationship between the eigenvalues and the matching number. In addition to these results, we present new results involving this family and the Chinese Postman Problem and a relationship between eigenvalues and edge-connectivity in regular graphs.
This is partly joint work with Sebastian M. Cioaba and Doulgas B. West.

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

The lack of national studies of the health effects of long-term exposure to ambient PM and its chemical components determining the PM toxicity represents a major evidence gap for the implementation of more effective air quality interventions. The US Environmental Protection Agency (EPA) is calling for research to explain heterogeneity in health responses to air pollutants that might be explained by the compositional differences in the pollution mixtures or sources or other factors.

We have developed Bayesian spatially varying coefficient regression models to estimate long-term effects of PM_2.5 on mortality while identifying the chemical composition that modifies the health effects. We will use spatio-temporal variation in health outcomes and exposure to estimate: 1) spatially varying health risks associated with long-term exposure to PM_2.5; and; 2) effect modification by PM_2.5 constituents. Our models will account for spatial misalignment of the data and uncertainty in the estimation of PM_2.5 chemical components.

We will apply our model to the Medicare Cohort Air Pollution Study (MCAPS) which includes 7.9 million Medicare enrollees followed for the period of 2000-2006 in the Eastern part of the US. We will use PM_2.5 data from 518 monitoring stations and PM_2.5 chemical components data from 241 monitors located in the Eastern region of the US. 

Hilbert scheme of points에 대한 정리들과 예들을 소개하고, Hilbert scheme의 scheme structre를 국소적으로는 어떻게 결정하는지에 대한, 즉 Deformation과 관련된 구체적인 계산을 보여줍니다.

In 1970's Vinberg proved the criterion for a Zariski dense subgroup generated by reflections to be definable over A where A is an integrally closed Noetherian ring in a algebraically closed field F. In this talk we state the criterion for a Zariski dense subgroup generated by reflections to be definable over A. By Borel and Benoist, Zariski density is not needed in the criterion for a reflection group which divides an irreducible properly convex set of the real projective sphere Sn. In this talk, we are not planning to give any proof. We will rather focus on explaining the definitions and results. Finally we will apply the criterion to compute all the integral representations of some hyperbolic n-simplex reflection groups which comes from the deformation space of convex real projective strucures. If time permits, we will also expalin how integral representations of some other hyperbolic polyhedral reflection groups such as triangular prismatic and cubical groups can be computed.

The aim of this talk is to give a first definition of "affine scheme" to people interested in algebraic geometry and its applications but who, have never tried to understand this language. We will briefly recall the definitions of category, functor and representable functor and...surprisingly this will be enough! Hopefully at the end of the talk we will define (affine) group schemes. In this talk we will assume that the audience is familiar with some very basic notions of commutative algebra.

To be announced

Hilbert scheme of points에 대한 정리들과 예들을 소개하고, Hilbert scheme의 scheme structre를 국소적으로는 어떻게 결정하는지에 대한, 즉 Deformation과 관련된 구체적인 계산을 보여줍니다.

A graph G is called perfect if for every induced subgraph H of G, the chromatic number and the clique number of H are equal. After the recent proof of the Strong Perfect Graph Theorem, and the discovery of a polynomial-time recognition algorithm, the central remaining open question about perfect graphs is finding a combinatorial polynomial-time coloring algorithm. (There is a polynomial-time algorithm known, using the ellipsoid method). Recently, we were able to find such an algorithm for a certain class of perfect graphs, that includes all perfect graphs admitting no balanced skew-partition. The algorithm is based on finding special “extremal” decompositions in such graphs; we also use the idea of “trigraphs”.
This is joint work with Nicolas Trotignon, Theophile Trunck and Kristina Vuskovic.

It is shown that for each t, there is a separator of size $O(t\sqrt{n})$ in any n-vertex graph G with no K_t-minor.

This settles a conjecture of Alon, Seymour and Thomas (J. Amer. Math. Soc., 1990 and STOC’90), and generalizes a result of Djidjev (1981), and Gilbert, Hutchinson and Tarjan (J. Algorithm, 1984), independently, who proved that every graph with n vertices and genus g has a separator of order $O(\sqrt{gn})$, because K_t has genus Ω(t²).
Joint work with Bruce Reed.

A tournament is a digraph obtained from a complete graph by directing its edges, and colouring a tournament means partitioning its vertex set into acyclic subsets (acyclic means the subdigraph induced on the subset has no directed cycles). This concept is quite like that for graph-colouring, but different. For instance, there are some tournaments H such that every tournament not containing H as a subdigraph has bounded chromatic number. We call them heroes; for example, all tournaments with at most four vertices are heroes.
It turns out to be a fun problem to figure out exactly which tournaments are heroes. We have recently managed to do this, in joint work with Berger, Choromanski, Chudnovsky, Fox, Loebl, Scott and Thomassé, and this talk is about the solution.

One of the main interest in algebraic number theory is to study Galois groups over number fields (and local fields) and their representations. In this talk, I will give an introduction to Galois representations obtained from elliptic curves or modular forms, and their relations. 

Modular curves as moduli spaces of elliptic curves with some additional structures provide an important tool for studying arithmetic of elliptic curves. In this talk we will explain how rational points of modular curves can be applied to the problem of determining the torsion subgroups and ranks of elliptic curves over number fields.

Hilbert scheme of points에 대한 정리들과 예들을 소개하고, Hilbert scheme의 scheme structre를 국소적으로는 어떻게 결정하는지에 대한, 즉 Deformation과 관련된 구체적인 계산을 보여줍니다.