곡면이 동그랗기 위한 충분조건은 적어도 14가지가 있다.
이 강연에서는 이들 중에서 평균곡률(mean curvature)을 이용한 충분조건에 관하여 옛 결과와 최근 결과를 알아볼 예정이다.

We discuss the Calderon problem: "Is it possible to determine the electrical conductivity inside a domain from the boundary voltage and current measurements?" This inverse problem is applied in the medical imaging. This introductory talk will give you a look at the famous Sylvester-Uhlmann’s method to prove the uniqueness of the Calderon problem. We also consider the non-uniqueness result, which is related with "Harry Potter's invisibility cloak".

A permutation tableau is a relatively new combinatorial object introduced by Postnikov in his study of totally nonnegative Grassmanian. As one can guess from its name, permutation tableaux are in bijection with permutations. Surprisingly, there is also a connection between permutation tableaux and a statistical physics model called PASEP (partially asymmetric exclusion process). In this talk, we study some combinatorial properties of permutation tableaux. One of our result is a sign-imbalace formula for permutation tableaux which is very similar to the sign-imbalace formula for standard Young tableaux conjectured by Stanley.

Meshless methods  (Meshfree methods, PUFEM, GFEM, XFEM) have several advantages over the conventional finite element method.Their flexibility and wide applicability have gained attention from scientists and engineers to this very dynamic research area. Recently, the author introduced one of most flexible closed form smooth partition of unity, named the generalized product partition of unity, whose PU functions have flat-top. Using the generalized product PU, we are able to construct  patchwise uniform polynomial reproducing particle (RPP) shape functions with Kronecker   delta property. The author also constructed local approximation functions that can handle  the Dirichlet BC for the thin elastic  plates, known as the Kirchhoff plate models.

Using a pair of 3-folds (typically blow-ups of Fano 3-folds), one can at once construct a Calabi-Yau 3-fold and a G_2 manifold, which are very different mathematical objects. We will decribe the methods and discuss some examples. 

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

In this talk, we are going to discuss  the homogenization process on self-organizing materials or information, and to find out the effective (or averaged) partial differential equations describing the first order approximation through filtering out the small oscillations occurred by inhomogeneous distribution of materials or information. One simple example is when two conductors with different conductivity distributed periodically on the plane with small periodicity. One of the interesting questions is what is the averaged effective conductivity. We are going to discuss Viscosity Method developed recently and to compare it with well known Energy method.

2차원 또는 3차원 쌍곡 공간에 살고 있는 다면체로는 어떠한 것들이 있는지 알아보고, 이러한 다면체를 이용하여 쌍곡 다양체를 만들어 보자.

Topological spaces with torus $T^n=(S^1)^n$ actions are very interesting objects, which have been studied for over a century as an important sub-branch of equivariant topology. A quasitoric manifold is determined by two conditions: the $T^n$-action locally looks like the standard $T^n$-representation in $\mathbb{C}^n$, and that the orbit space is combinatorially a simple convex polytope $P^n$. Both conditions are satisfied for the torus action on a non-singular projective toric variety, but there are examples of quasitoric manifolds which are not toric varieties. In this talk, I introduce some examples and properties of quasitoric manifolds. If time allows, I explain the relationship among quasitoric manifolds, moment angle manifolds and small covers.

In this talk, we will see the height of a Kummer surface of special type is determined by the associated abelian surface using the Frobenius diagram of the Kummer fibration. 

Given a fixed $p \neq 2$, we prove a simple and effective characterization of all radial mutipliers
of $\mathcal{F}L^p(\mathbb{R}^d)$, provided that the dimension $d$ is sufficiently large. The method also yields new $L^q$ space-time regularity results for solutions of the wave equation in high dimensions.

In 1989, Stephenson and Zelen derived an elegant formula for the information I_ab contained in all possible paths between two nodes a and b in a network, which is described as follows. Given a finite weighted graph G and its Laplacian matrix L, the combinatorial Green’s function , of G is the inverse of L+J, where J is the all 1’s matrix. Then, it was shown that I_ab=(g_aa+g_bb-2g_ab)^-1, where g_ij is the (i,j)-th entry of . In this talk, we prove an intriguing combinatorial formula for I_ab:

,

where  is the complexity, or tree-number, of G, and G_a*b is obtained from G by identifying two vertices a and b. We will discuss several implications of this new formula, including the equivalence of I_ab and the effective conductance between two nodes in electrical networks.

In the history of number theory, Diophantine equations are one of the main stream. A Diophantine equation, named after Hellenistic mathematician Diophantus, is an indeterminate polynomial equation which allows the integral solutions only. There are many questions related to Diophantine equations. You can ask whether solutions exist, how to find them, and so on. In this seminar, I introduce a basic Diophantine problem, called "the sum of two squares". And then I introduce some concepts which are the building block of algebraic number theory that follow from Diophantine problem. And, so far as time permits, I give partial answer for questions related to Diophantine equations.

We have enumerated all the prime theta-curves
and handcuff graphs with up to 7 crossings before.

We can composite many spatial graphs by using ``connected sum'' of them.
However, for spatial graphs, ``connected sum'' is not unique.

In this talk, we enumerate non-prime theta-curves and handcuff graphs
with up to seven crossings by using algebraic tangles and
non-prime basic theta-polyhedra.

We study the relative full-flag Hilbert scheme of points on the family of curves, parameterizing chains of subschemes, containing a node. We will prove that the relative full flag Hilbert scheme is normal with locally complete intersection singularities. 

One of the most fundamental properties of a connected graph is the existence of a spanning tree. Also the number τ(G) of spanning trees is an important graph invariant. It plays a crucial role in Kirchhoff’s classical theory of electrical networks, for example in computing driving point resistances. More recently, τ(G) is one of the values of the Tutte polynomial which now plays a central role in statistical mechanics. So are a(G), the number of acyclic orientations, and c(G), the number of orientations in which every edge is in a directed cycle. As a first step towards convexity properties of the Tutte polynomial, Merino and Welsh conjectured that

for every loopless and bridgeless multigraph G. We shall here prove that τ(G) ≤ c(G) for all loopless and bridgeless multigraphs with n vertices and at least 4n edges and that τ(G) ≤ a(G) for all loopless multigraphs with n vertices and at most 16n/15 edges. We also verify the conjecture for cubic graphs (which are in between these two bounds).

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

A rendezvous number for a metric space M is a number a such that, for every finite subset Q of M, there is an element p in M, such that the average distance from p to the elements in Q is a.

O. Gross showed in 1964 that every compact connected metric space has precisely one rendezvous number. This result is a consequence of von Neumann’s min-max theorem in game theory.

In an article in the American Math. Monthly 93(1986) 260-275, J. Cleary and A. A. Morris asked if a (more) elementary proof of Gross’ result exists.

In this talk I shall formulate a min-max result for real matrices which immediately implies these results of Gross and von Neumann.

The proof is easy and involves only mathematical induction.

Rational homology projective planes are normal projective surfaces having same Betti numbers with the complex projective plane. The first half of the talk will be devoted to the introduction to the topic: examples, relations with other classification problems. For the remaining part of the talk, I will present the recent results on the algebraic Montgomery-Yang problem. 

컴퓨터의 발명은 인간에게 많은 양의 수학적 계산을 짧은 시간에 해낼 수 있는 능력을 가져다주었다.  이러한 변화는, 주어진 문제의 답의 존재성에 더 관심을 두는 수학의 전통적인 접근법과는 다르게 답을 제한된 계산 자원 (시간, 메모리 등)을 이용하여 효율적으로 계산하는 방법에 대한 새로운 문제의 중요성을 대두시키게 된다.

이 강연에서는 효율적으로 계산 가능한 문제 (P)와 효율적으로 검증가능한 문제 (NP)에 대해 알아보고, approximation algorithm, randomized algorithm 등 효율적인 계산이 어려운 문제들에 대한 접근법과 암호론 등에서의 응용에 대해 알아본다.

We discuss how to efficiently compute shortest and approximate shortest paths in graphs, thereby focussing on shortest path query processing. The algorithm computing (approximate) shortest path queries is allowed to access a pre-computed data structure (often called distance oracle) in order to improve the query time. Several questions regarding such data structures are of particular interest: How can they be computed efficiently? What amount of storage is necessary? How much improvement of the query time is possible? How good is the approximation quality of the query result? What are the tradeoffs between pre-computation time, storage, query time, and approximation quality?

For general dense graphs, the tradeoff between the storage requirement and the approximation quality is known up to constant factors. We discuss both the lower and the upper bound (by Thorup and Zwick). For specific types of sparse graphs, however, the exact tradeoff is not known; the general tradeoff can be improved: there are special data structures of a certain size that enable query algorithms to return distances of higher quality than what the general tradeoff would predict. We outline the state of the art of distance oracles for planar graphs and other classes of sparse graphs. We then prove that this improvement for some classes of sparse graphs cannot be extended to all sparse graphs: there is a three-way relationship between space, query time, and approximation quality for general sparse graphs. If time permits, we also cover space- and time-efficient data structures for certain complex networks with power-law degree sequences.

Joint work with Wei Chen, Shinichi Honiden, Michael Houle, Ken-ichi Kawarabayashi, Shang-Hua Teng, Elad Verbin, Yajun Wang, Martin Wolff, and Wei Yu.

A J-holomorphic curve is a map from a Riemann surface to an almost complex manifold (M,J) whose differential preserves almost complex structures. The concept of J-holomorphic curves is a powerful tool to study symplectic manifolds. A symplectic manifold always admits an almost complex structure J and J can be chosen to be "tamed" by the symplectic structure. In this case, J-holomorphic curves behave well and we can study symplectic manifolds by studying J-holomorphic curves. In this talk I will explain some results obtained by using J-holomorphic curves.

 The constant demand for increasingly accurate, efficient, and robust numerical methods, which can handle acoustic, elastodynamic and electromagnetic wave propagations in unbouded domains, spurs the search for improvements in artificial boundary conditions. In the last decade, the perfectly matched layer (PML) approach has proved a flexible and accurate method for the simulation of waves in unbounded media. Standard PML formulations, however, usually require wave equations stated in their standard second-order form to be reformulated as firstorder systems, thereby introducing many additional unknowns. To circumvent this cumbersome and somewhat expensive step we propose instead a simple PML formulation directly in its second-order form in 3D. Our formulation requires fewer auxiliary unknowns than previous formulations. Starting from a high-order local nonreflecting boundary condition (NRBC) for single scattering, we derive a local NRBC for time-dependent multiple scattering problems, which is completely local both in space and time. To do so, we first develop a high order exterior evaluation formula for a purely outgoing wave field, given its values and those of certain auxiliary functions needed for the local NRBC on the artificial boundary. By combining that evaluation formula with the decomposition of the total scattered field into purely outgoing contributions, we obtain the first exact, completely local, NRBC for time-dependent multiple scattering. The accuracy, stability and efficiency of this new local NRBC is evaluated by coupling it to standard finite element or finite difference methods.

In this talk, I present an innovative nonparametric Bayes methodology for ex-ibly characterizing the relationship between a continuous response and multiple predictors.

Goals are (1) to estimate the conditional response distribution addressing the distributional changes across the predictor space, and (2) to identify important predictors for the response distribution change both within local regions and globally. We rst introduce the probit stick-breaking process (PSBP) as a prior for an uncountable collection of predictor-dependent random distributions and propose a PSBP mixture (PSBPM) of normal regressions for mod-eling the conditional distributions. A global variable selection structure is incorporated to discard unimportant predictors, while allowing estimation of posterior inclusion probabili-

ties. An ecient stochastic search sampling algorithm is proposed for posterior computation.

The methods are illustrated through simulation and applied to an epidemiologic study.

1. 금번 금융위기에 대한 금융공학적 접근:

주택관련 파생상품을 중심으로 발생된 2008년 금융위기의 발생 원인과 해결 과정에서 금융공학의 역할을 분석하고, 금융공학의 미래에 대해 전망한다.

2. 금융시장에서 수학의 중요성 확대:

금융시장의  다양한 분석 방법과 최근 조류에 대한 조망을 통해 금융시장에서 수학이 필요한 이유와 비중 확대 가능성을 제시한다.

There are some important PDEs in fluid dynamics, such as heat equation, wave equation, and Stokes equation, etc. Numerical methods to solve those PDEs has been developed and now there are two kind of method widely used. Finite Difference and Finite element method.  Finite element method can be described as follows. first, formulate given equations as weak equations, and determine the space in which we seek for original solution. second, find admissible finite dimensional function space in which we seek for numerical solution. third, discretize weak formulations and reduce original problem as a set of linear equations

I will give a brief introduction to invariant metrics (the Kobayashi, Carath\'eodory, Bergman, and Sibony metric) and explain how the asymptotic behavior of the metrics near the boundary of a domain is related to the geometry of the boundary. The talk will be accessible to graduate students.

I will give an introduction to Stein complex manifolds. I will describe basic examples and non examples for domains in complex Euclidean space. Trickier examples arise from the interplay with isolated singularities of surfaces, an interesting topic in its own. I will assume familiarity with smooth manifolds, and interest in the field of several complex variables. 

In this talk, I will introduce high-dimensional data analysis and related problems. Traditionally it is considered as a statistical problem, but due to its innate difficulty, often described as the curse of dimensionality, it produces many challenging and interesting mathematical problems and more and more mathematicians are interested in its geometry and analysis, considering data sets as discrete or sampled continuous geometric structures embedded in high-dimensional spaces. With such view point, I'll explain Laplacian, eigenfunctions and heat equation on data sets and graphs and talk about their applications.