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

We will define quasitoric orbifolds, describe their topological
invariants. We will discuss almost complex structures and blowdowns
on four dimensional quasitoric orbifolds and a McKay correspondence
for these.

We will discuss basic facts about differentiable orbifolds and
intorduce Chen-Ruan cohomology groups and Mckay correspondence.

Many interesting works on topological algebras are best understood in view of algebraic topology or diffrential geometry. Since Gelfand-Naimark's celebrated characterization of commutative C*-algebras as a dual of locally compact Hausdorff space, the category of non-commutative C*-algebras has been regarded as "non-commutative locally compact Hausdorff spaces". In this talk, he will introduce so called non-commutative (algebraic) topology and (differential) geometry relying on his own appreciations of the subjects. 

The notion of a matroid was introduced by H. Whitney in 1935 as an abstraction of “linear independence” in a vector space. In 1960’s, J. Edmonds found that matroids are closely related to an efficient algorithm. After then, many researchers have studied and extended Matroid Theory. In 2007, S. Fujishige, G. A. Koshevoy, and Y. Sano introduced cg-matroids as an generalization of matroids. They defined a matroid-like structure on an abstract convex geometry, where the notion of an abstract convex geometry was introduced by P. H. Edelman and R. E. Jamison in 1985. In this talk, I will introduce the theory of cg-matroids. In particular, I will talk about the definition and some examples of cg-matroids, some combinatorial properties and characterization of cg-matroids, and the relationship between cg-matroids and the greedy algorithm.

In this talk I will introduce to two topological objects, that are topological configuration spaces and discrete configuration spaces, and Discrete Morse theory, which is one of combinatorial versions of Morse theory. We will discuss relations between topological configuration space and discrete configuration spaces, and applications of Discrete Morse theory to studing discrete configuration spaces.

The only possible fibration on a K3 surface is an elliptic fibration, with base P^1. Lagrangian fibrations on holomorphic symplectic manifolds are higher-dimensional generalizations of elliptic K3 surfaces. In this talk I will describe some results about Lagrangian fibrations concerning their bases, discriminant loci, singular fibres, and classifying maps. 

Recent brain research relies heavily on connectionism since connection exists universally in the brain, for example, between neuron and neurons, between region and regions, between brains and cultures. To examine different levels of connection within brain, various neuroimaging techniques have been suggested to show functional brain connection. Anatomic connection is provided by diffusion tensor imaging and functional connection is dealt by functional MRI. These techniques have been used for diagnosis and exploration of diseases and also for understanding brain and interactions. In this presentation, we will review advanced methods for brain connectivity in the respect of anatomic connection, functional connection from neuronal level to social brain level.

세상 모든 건은 서로 연결되어 있다는 간단한 생각에서 출발한 복잡계에 대한 연구는 21C에 들어와 물리학의 중요한 주제로 떠오르고 있다. 복잡계연구에서는 여러 분야의 서로 다른 많은 수의 객체(노드)들 간의 복잡한 상요작용(링크)이 얽혀진 네트워크를 분석함으로써 복잡계의 구조의 성질 및 그 안에서 일어나는 여러 협동현상들을 이해하려 하고 있다. 이번 콜로퀴엄에서는 복잡계 네트워크 과학을 소개하고 IT분야 Google의 전략과 미래인터넷연구, 시스템생물학을 통한 신약개발 노력, 18대 국회의원의 네트워크 분석, 도로교통망의 체증해소 문제, 컴퓨터 바이러스와 AIDS 등의 전염병 확산방지대책등 최근의 복잡계 연구동향을 간략히(?) 살펴보고자 한다.

Matrix analogues of embedded surfaces, as well as discrete versions of curvature
and the Euler characteristic will be presented, and the new notions, related to a novel algebraic formulation
of classical differential geometry, will be illustrated by a large class of examples.

We shall construct infinitesimal deformations of a complex surface with boundary, which is the complement of a tubular neighborhood of a special linear chain of smooth rational curves. We show that the deformation induces a non-trivial deformation of the complex structure on some divisor on Y. 

  장 소 : 자연과학동(E6-1) #3433

 일 정 : 2010. 5. 3(월) 15:00~17:30

  1. 권오상 박사 (15:00~15:15)

    - Title : Existence of standing wave for Nonlinear Schrodinger equation

  2. 김정애 박사 (15:15~15:30)

   - Title : Nonlinear viscoelastic equation

  3. 신동화 박사 (15:30~15:45)
   - Title : Diophantine equations and modular forms

  4. 윤석배 박사 (15:45~16:00)

    - Title : An introduction to Kinetic equations

  5. 이완석 박사 (16:00~16:15)

    - Title : On the classification of non-normal del Pezzo varieties

  6. 임병화 박사 (16:15~16:30)

    - Title : Firm Size Growth Effect on the Risk-Taking Behavior: BSDE Approaches

  7. 장광성 박사 (16:30~16:45)

    - Title : Extraction immersed finite element method for Stokes flow with singular

               forces

  8. 정의진 박사 (16:45~17:00)

    - Title : Classification problems in symbolic dynamics

  9. 주항연 박사 (17:00~17:15)

    - Title : On Homoclinic points, Recurrences and Chain recurrences of volume-preserving

               diffeomorphisms without genericity

  10. Zribi, Habib 박사 (17:15~17:30)

    - Title : Small Perturbation of an Interface: Asymptotic Formulas and Reconstruction

             of Small Interface Changes

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.