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.

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

In this talk, a dynamic algorithm to predict the pricing of bond options using actual data set of gilts will be considered. The pre-measured interest rate traces lose the meaning of interest rate prediction because the actual distribution is not equal to the normal distribution which is used for bond option models in general. Thus, to obtain the interest rate dynamics, we need the cumulative distribution function (CDF) of actual data of gilts using numerical methods. In addition, Pricing models of Bond Option with jumps will be discussed.

Latent Dirichlet Allocation (LDA) and Hierarchical Dirichlet Processes

(HDP) have become popular models for discovering latent semantics from

text corpora. I will first start this talk with what LDA and HDP are and

how they are used in common text analysis tasks. Then, I will present

three recent papers from our research group that extend LDA and HDP to

analyze three different text corpora: online reviews, news articles, and

conference proceedings. With the online reviews, we propose a variant of

LDA called Aspect and Sentiment Unification Model (ASUM) to analyze topics

and sentiments jointly in an unsupervised fashion. With the news articles,

we use LDA to generate topic chains to model temporal patterns of similar

topics. With the conference proceedings, we propose a variant of HDP

called distant dependent Chinese Restaurant Franchise (ddCRF) to discover

how new topics emerge through time.

Random walks in random scenery (RWRS) are processes where a random
walker collects a random reward (or scenery) at each site it visits.
If it visits a site multiple times, it collects the same reward many
times thus leading to correlations in this partial sum process. Cohen

as their scaling limits. The limiting processes can be written as
stochastic integrals with random kernels, and the processes have

We consider a modification in which a sign associated to the reward
(scenery) alternates upon successive visits.  The resulting process is

are also stochastic integrals with quite simple random kernels of
indicator type. Our results complement the above results in that the

In 2007, families of new examples of homogeneous domains were presented
by K.H. Lee - homogeneous in the sense that the action by their J - holomor-
phic automorphism group is transitive. To the surprise of many experts, such
examples are almost-complex but not complex. Such examples were formerly
expected to be non-existent, or if they existed, they should be sporadic and
very rare, such as nitely many. But not only are there innitely many such
examples, but they form a continuous family when the dimension (of the man-
ifold) is 6 or higher. So such homogeneous almost-complex (but not complex),
which we call the models, have emerged as a research object that requires careful
investigation.
In this talk, we introduce the uniformization theorem in Riemann surface and
the dierential geometry of the standard unit ball. In this viewpoint, we discuss
the dierential geometric characterization of the models which are mentioned
above.

2000년대 이후 우리 사회에는 무선 통신을 위한 인프라가 구축되었으며 많은 무선 통신 기기가 보급되었다. 그 이면에는 이를 뒷받침할만한 통신 기술의 발전이 있었으며 시스템의 성능을 분석하는데 있어서 수학이 큰 역할을 하였다. 이번 발표에서는 우선 시스템의 성능을 분석하는 대표적인 방법인 큐잉 이론에 대하여 알아본다. 큐잉 이론은 전화 교환망에 대한 성능 분석을 시작으로 현재 존재하는 거의 모든 통신 시스템의 성능을 분석하는데에 쓰여 왔다. 또한, 큐잉 이론을 이용하여 무선 통신에서 가장 간단한 형태인 일대일 통신의 성능을 큐잉 이론을 적용하여 분석하는 방법에 대하여 소개하고 차세대 통신 기술로써 현재 많은 연구가 되고 있는 인지무선통신에 대해서 간단히 소개한다.

Given two graphs $G$ and $H$, the categorical product $G \times H$ has vertex set $V(G) \times V(H)$, and two vertices $(x,y)$ and $(x',y')$ are adjacent if $xx' \in E(G)$ and $yy' \in E(H)$. The famous Hedetniemi-Lovasz Conjecture asserts that teh chromatic number of $G \times H$ equals the minimum of $\chi(G)$ and $\chi(H)$. In this talk, I will sketch a proof of the fractional version of the conjecture, which says that the fractional chromatic number of $G \times H$ equals to the minimum of the fractional chromatic numbers of $G$ and $H$. This result is then used to prove a conjecture of Burr-Erdos-Lovasz on the chromatic Ramsey number of graphs.

In 1970s, B. C. Berndt has found a more general class of Eisenstein series and has computed 
transformation formulas for them. He also has converted these transformation formulas into 
transformation formulas for a large class of functions containing the classical Dedekind 
eta function. By this process, he has established a number of interesting infinite series identities, 
many of which are stated in the Notebooks of Ramanujan or are proved by many authors. 
In this talk, we introduce his work and, continuing his work, we prove three modular transformation 
formulas. Furthermore we give more general infinite series identities regarding hyperbolic 
functions and trigonometric functions. 

The pseudo-spectral (or spectral collocation) method is a numerical approximation method based on spectral basis functions for differential operators on a discretized domain specified with boundary conditions. Considered as the generalized discretized Fourier transformation, this method provides spectral accuracy with respect to
the number of discretization in approximating differentiable functions, therefore it is a powerful tool in numerical approximation for smooth enough solutions to PDE in accuracy and efficiency. In this talk, some applications of this method are presented, particularly highlighting how to properly treat multi-dimensional, unbounded,
singular properties in solving some nonlinear PDE that naturally arise in the course of mathematical modeling in physical science and engineering.

소셜 네트워크를 비롯한 현실 속의 많은 시스템들은 서로 긴밀히 연결되어 있는 복잡계 망의 구조를 갖는다. 복잡계 망에서의 정보 확산에 있어서, 특정 임계점을 넘어서면 정보가 전체 시스템으로 급격히 확산되는 상전이 현상이 관찰된다. 이 발표에서는 복잡계 망의 특성 및 정보 확산 모델을 소개한다. 에르되시-레니 모델부터 일반화된 무작위 그래프까지 다양한 확률적 모델들을 통한 네트워크 분석 방법을 알아보고 관련된 연구 주제 및 기대 성과를 생각해 본다.

Algebraic quantum field theory aims at providing a mathematical description of the structure of quantum field theories. In this talk I will introduce the basic notions of quantum field theory and QFT algebra, sketch some relevant classical ideas - such as homological and homotopy algebras, basic deformation theory, etc, for a general audience.

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

In this talk, I will introduce the notions of QFT algebra and corresponding Quantum Homotopy Invariants. It will be a quantun version of the first talk. 

BK21 연수연구원 세미나 (2011.4.14(목), 16:00~17:30, 자연과학동 3433호)

1. 권오상

-제목 : Existence of solutions for elliptic partial differential equations

-Abstract

: In this talk, I will briefly review the basic concepts of the calculus of variations and Lyapunov-Schmidt reduction method, and prove the existence of solutions for some elliptic partial differential equations using Lyapunov-Schmidt reduction method.

2. 박혜인

-제목 : Heat kernel estimates for Parabolic operators

-Abstract

: We study the relation between parabolic operators and markov processes. Also we establish the heat kernel estimates for the parabolic operators.

3. 정재환

-제목 : Number of zeros and Heat equation in the real line

-Abstract

 : For a solution to the one dimensional heat equation, number of zeros of the solution does not increase with time; we will discuss the nonincreasing property and its applications. Long time asymptotics of zeros will be also mentioned.

4. 곽민석

-제목 : Optimal Portfolio, Consumption and Retirement Decision under a Preference Change

-Abstract 

: We consider an optimal portfolio, consumption and retirement decision problem in which an economic agent can determine the discretionary stopping time as a retirement time.

It is assumed that the agent's coefficient of relative risk aversion becomes higher after retirement.

Under a CRRA utility function, we obtain the optimal policies in closed-forms using martingale methods and variational inequality methods.

We analyze the properties of the optimal policies with numerical examples.

5. 조윤형

-제목 : TBA

One version of Kakeya's problem asks for the smallest convex garden in which a ladder of length one can be rotated fully.  The answer to this question was shown to be the equilateral triangle of height one by Pal in 1920.

We consider the following generalization: Given a (not necessarily finite) family F of line segments in the plane, what is the smallest convex figure K such that every segment in F can be translated to lie in K?

We show that as in the classic case, the answer can always be chosen to be a triangle. We also give an O(n log n) time algorithm to compute such an optimal triangle if F consists of n line segments.

Joint work with Sang Won Bae, Hee-Kap Ahn, Joachim Gudmundsson, Takeshi Tokuyama, and Antoine Vigneron.

We consider two and three dimensional chemotaxis system coupled with the fluid equations.

This model describes the dynamics of oxygen-driven bacteria which is living in the fluids.

We consider the local existence and global existence of the solution.

Based on various experiences of speaker as teaching assistants, in this talk, we first describe a 'good/bad‘ teaching assistant in Mathematics for students, professors, Head TAs, and oneself. After reviewing some specific cases, in order to be a good TA, let us consider what to do or what not to do.

In this talk, I will explain how to construct the quantum invariants for classical singularity theory via constructing the virtual cycle on moduli spaces of $W$-curves. Those invariants are closely related to the study of mirror symmetry and integrable hierarchy.

BK21 연수연구원 세미나 (2011.4.6(수), 16:00~17:30, 자연과학동 3433호)

1. 신동화

-제목 : Class number one problem

-Abstract

: We revisit the class number one problem done by Heegner and Stark, and present a new proof by making use of Shimura's reciprocity law.

2. 최영준

-제목 : The differential geometric models in almost complex manifolds and the negatively curved Kähler manifolds

-Abstract

: In this talk, we introduce the differential geometric Models in Almost Complex Manifolds and discuss their differential geometric characterization theorem. We also introduce the classical problem related the uniformization theorem, in particular the Negatively curved Kähler manifold.

3. 박선정

-제목 : Topological classification of quasitoric manifolds

-Abstract

: A quasitoric manifold is a smooth manifold with locally standard torus action whose orbit space is a simple convex polytope. In this talk, I will explain a cohomological rigidity problem and give some affirmative results, that is, some quasitoric manifolds are classified by their integral cohomology rings up to homeomorphism

Although singularities of degree 1 or -1 appear in the energy minimizers for harmonic maps or Oseen-Frank energy, there are many situations where singulairties of degree 1/2 or -1/2 have been observed. But there is no mathematical frame work to prove such singularities. In a joint with P. Bauman and D. Phillips, we use Landau-de Gennes Energy to prove that such singularities do exist. In the proof, we employ a famous well-known theory developed by Bethuel, Brezis, and Helein.

The study of singularities (such as ADE type singularities) has a long history in mathematics, and has also appeared in physics.  In this talk, I will explain the basic concepts in quantum singularity theory. For example, $W$-curves, Witten equation, moduli spaces and  the relation with $r$-spin curve theory, and integrable hierarchy.

20세기에야 비로소 독립된 분야로 확립된 게임이론은, 수학에 그 큰 빚을 지고 있다. 경제주체들의 행동을 설명하고자 했던 이 수학적 시도는, 불규칙적이고 혼란스러워 보이는 인간행동에 수학적 법칙과 질서를 부여한다.
이 세미나에서는, 누구에게나 친숙한 Nash Equilibrium을 출발점으로 게임이론의 중심화두 중 하나인 "균형"에 대해 살펴보고자 한다. 특히, Axiomatic Bargaining Game Theory의 흥미로운 예들을 통해, 수학적 균형점이 가지는 사회과학적 함의에 대해 함께 생각해 보자.