In this talk, we consider mathematically and computationally optimal control problems for stochastic partial differential equations under the Neumann boundary condition. The control objective is to minimize the expectation of a cost functional, and the control is of the deterministic, boundary value type. Mathematically, we prove the existence of an optimal solution and a Lagrange multiplier; we represent the input data in their Karhunen-Loµeve (K-L) expansions and deduce the deterministic optimality system of equations. Computationally, we approximate the finite element solution of the optimality system and estimate its error through the discretizations of both the probability space and the spatial space.

Sophus Lie developed the theory of Lie groups and their Lie algebras. The theory of semisimple Lie algebra is useful in many parts of mathematics and physics. Many results of Lie algebra are corresponding the thing of Lie group such as representation theory and the classification problem. The semisimple Lie algebras over C are in one to one correspondence with compact and simply connected Lie groups. In the period 1888 - 1894 much of the structure of Lie algebras over C was developed, including W. Killing's discovery of "Killing form" in E. Cartan's thesis. And, Weyl's Theorem was orginally proven using integration on compact Lie groups. An algebraic proof of Weyl's theorem was found in 1935 by Casimir and van der Waerden.
The semisimple Lie algebras over C were first classified by Wilhelm Killing though his proof lacked rigor. His proof was made rigorous by Elie Cartan (1894) in his Ph.D. thesis, who also classified semisimple real Lie algebras. This was subsequently refined, and the present classification by Dynkin diagrams was given by then 22-year old Eugene Dynkin in 1947. Some minor modifications have been made (notably by J. P. Serre), but the proof is unchanged in its essentials.
In this talk, we discuss the classification problem of finite dimensional semisimple Lie algebras over C. And we give some relevant concepts of classification problem such as Dynkin diagram and root systems and some algebraic property of Lie algebras.

We will consider a program launched by C. S. Morawetz about the existence of weak solutions to two-dimensional, steady, irrotational transonic flow equations. Among other things we will consider an analysis of the boundary conditions for the viscous problem and an analysis of the entropy functions which are generated by a Tricomi-like mixed-type equation.

수리과학과에는 30여명의 교수님이 계시고 교수님들의 연구 분야는 모두 다른데 각 교수님의 연구 분야를 자세히 들을 기회가 부족했기 때문에 이번 학기부터 대학원 진학을 희망하는 학부생과 지도교수를 정하지 않은 대학원 신입생에게 연구 분야 소개 행사를 시작했습니다.

11월 23일에는 Weidong Ruan 교수님이 소개하십니다.

Our department has about 30 professors with various research areas but it was rather difficult to listen to research areas in detail. Starting from this semester, we have events to introduce research areas to undergraduate students who are interested in graduate schools or the first year graduate students who did not yet decide advisors.  

On Nov. 23, Prof. Weidong Ruan will introduce his research area.

Laguerre histories are certain colored Motzkin paths with some weight for each elementary steps. In this talk, we study two famous bijections between permutations and Laguerre histories, made by Francon-Viennot and Foata-Zeilberger. This two bijections are enable us to give permutations as combinatorial interpretations of continued fractions. The former is associated in linear statistics and the latter be in cyclic statistics. Using two mappings, we are able to make various results about several statistics of permutations.

The $G_{2}$-manifolds are Riemannian manifolds with the speical holonomy group $G_{2}$. Here  $G_{2}$ is one of the exceptional Lie groups, which is the automorphism group of octonions. In this talk, we introduce the geometry of $G_{2}$-manifolds and the duality theory motivated from M-theory in Physics. We also discuss recent progress on the geometry from the analogous point of view to the symplectic geometry. 

To be announced

학사과정 무학과 학생의 학과 예비신청에 도움을 주기 위한 수리과학과 설명회가 열립니다. 많은 참석 바랍니다.

(피자제공)

I will discuss some computational problems that arise naturally and frequently in algebraic geometry. Esepcially I will explain how to formulate correct notion of limits of algebraic varieties, algorithms for computing the limits, and the cost of such computations. I will give various applications to several different areas, including a recent result in birational geometry of moduli space of stable curves.

Since 1970s, the study of topological spaces with torus actions has played an important role in various areas of mathematics, such as algebraic topology, algebraic geometry, convex geometry, symplectic geometry, combinatorics, and commutative algebra. As it becomes widespread, recently a field of activity called ``Toric topology'' has emerged. Davis and Januszkiewicz first introduced a certain universal $T^m$-manifold whose orbit space is a combinatorial simple polytope P. This manifold has the following universal property : for every quasitoric manifold $M^{2n} \to P$ there is a principal $T^{m-n}$-bundle $Z_P \to M^{2n}$ whose composite map is the orbit map for $Z_P$. This space $Z_P$ is later called "moment angle manifold". Buchstaber and Taras Panov extended this space to the case of arbitrary simplicial complex, and they named those spaces the "moment-angle complexes". Recently, Zhi Lu and Taras Panov extended the construction of moment-angle complexes to simplicial posets by using some categorical language.
In this talk, we discuss some facts about simplicial posets and some backgrounds about the homological algebra related to the face ring of simplicial posets. And we investigate the study of the topology of the moment angle complexes. Finally we introduce some recent works and remaining questions.

The zero level set of a solution to the heat equation is considered. Under certain conditions on moments of the initial data, we will prove the zero level set in a ball converges to a specific graph as time goes to infinity. Solving a linear combination of the Hermite polynomials gives the graph and coefficients of the linear combination depend on moments of the initial data. Also the graphs to which the zero level set converges can be classified in some cases

Minimization of functionals related to Euler's elastica energy has a wide range of applications in computer vision and image processing. An issue is that a high order nonlinear partial differential equation needs to be solved and the conventional algorithm usually takes high computational cost. In this talk, we propose a fast and efficient numerical algorithm to solve minimization problems related to the Euler's elastica energy and show applications to variational image denoising, image inpainting, and image zooming. We reformulate the minimization problem as a constrained minimization problem, followed by an operator splitting method and relaxation. The proposed constrained minimization problem is solved by using an augmented Lagrangian approach. Numerical tests on real and synthetic cases are supplied to demonstrate the efficiency of our method.

The study of electric potentials on graphs has a close connection with probability theory. On distance-regular graphs the theory is particularly elegant, as known results concerning such graphs allow explicit calculations to be made. In this talk, I will introduce these topics and discuss some recent results that Jacobus Koolen and I have obtained, as well as some conjectures and open problems.

In this talk, I will discuss the existence of positive solutions of Hessian equations of Lane-Emden type.
In particular, I will explain the structure of solutions which are compared with the structure on Lane-Emden equations.

We obtain an almost sure version of a maximum limit theorem for stationary Gaussian random fields under some covariance conditions. As a by-product, we also obtain a weak convergence of the stationary Gaussian random field maximum, which is interesting independently.

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

One of the fundamental problems in 4-manifolds is to classify simply connected smooth (symplectic, complex) 4-manifolds. The classical invariants of a simply connected  4-manifold X are encoded by its intersection form Q_X, a unimodular symmetric bilinear pairing on H_2(X:Z).  M. Freedman showed that a simply connected closed 4-manifold is determined up to homeomorphism by Q_X. But the situation in the smooth (symplectic, complex) category is strikingly different. Hence it is an important question to know which unimodular, symmetric, bilinear integral forms are realized as the intersection form of a simply connected smooth (symplectic, complex) 4-manifold, and which simply connected smooth (symplectic, complex) 4-manifolds admit more than one smooth (symplectic, complex) structure. We call these geography problems of 4-manifolds.
Gauge theory - Donaldson theory and Seiberg-Witten theory - has been very successful in the geography problems. For example, S. Donaldson proved that the intersection form of a simply connected, definite, smooth 4-manifold is diagonalizable, and it has been known that most known simply connected irreducible smooth 4-manifolds with b_2^+ odd > 1admit infinitely many distinct smooth structures. Recently, there has also been a big progress in the case of b_2^+ = 1.
In this talks I’d like to review recent known results on the geography problems of 4-manifolds, in particular  in the case of b_2^+ = 1, in three levels - smooth, symplectic and complex category.  

To be announced

Since introduced by Wigner, random matrix theory has become a powerful tool in mathematical phyics. The subject now also plays important roles in various fileds such as combinatorics, probability theory, statistical physics, number theory, nuclear physics, game theory, wireless communication, etc. In this talk, I will explain some important results on random matrices such as Wigner semi-circle law and Dyson sine kernel. Some applications will also be introduced.

Freeness of a hyperplane arrangement is defined algebraically using module of derivations. I will discuss freeness of certain truncated affine Weyl arrangements, called the extended Catalan/Shi arrangements. I will also talk some enumerative corollaries.

I will talk about several results on geometric flow. In particular, I will talk about prescribed (Gaussian) curvature flow, CR Yamabe flow, and Q-curvature flow.

The cyclic sieving phenomenon is introduced by Reiner, Stanton and White in 2004. Since then this new topic attracts more and more attentions of researchers in recent years. In this talk we will introduce the cyclic sieving phenomenon and introduce some interesting old and new results. Finally we present a new development on constructing new cyclic sieving phenomena from new ones via elementary representation theory.

The Strichartz estimate has been extensively studied as a basic tool for nonlinear problems of dispersive equations.

In this talk we will brief on the Strichartz estimates for linear Schrödinger waves and apply them to the wellposedness and smoothing property of NLS.

The stable marriage problem is a classical matching problem. An input consists of the set of men, the set of women, and each person’s preference list that orders the members of the opposite sex according to the preference. The problem asks to find a stable matching, that is, a matching that contains no (man, woman) pair, each of which prefers the other to his/her current partner in the matching.

One of the practical extensions is to allow participants to use ties in preference lists and to exclude unacceptable persons from lists. In this variant, finding a stable matching of maximum size is NP-hard. In this talk, we give some of the approximability results on this problem.

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

We will survey how to improve computational efficiency in various pairing computations.

1900년 파리 국제수학자대회에서 힐버트(Hilbert)는 20세기 수학자들이 고민하고 연구해야할 문제 23개를 발표했습니다. 그 중에 14번째 문제는 본래 불변량 이론(Invariant Theory)의 문제였으나, 20세기에는 대수기하, 가환대수, 표현론 등의 분야와 관련하여 많은 연구 결과들이 나오게 되었습니다. 특히, 1959년 일본의 나가타(Nagata)는 14번째 문제에 대한 반례를 발견하였는데, 나가타가 증명 과정에서 사용한 핵심적인 아이디어는 이후 대수기하에서 매우 중요한 역할을 하게 됩니다. 이번 대학원생 세미나에서는 힐버트의 14번째 문제와 관련된 대수기하의 내용에 대해서 발표할 예정입니다.

If we consider strongly-pseudo convex CR manifolds with integrable CR structure, the transformation formula of the pseudo-Hermitian scalar curvatures satisfies a very special non-linear sub-elliptic PDE, which is called the CR Yamabe equation. In 1995, R. Schoen made use of this equation for the  characterization of the Heisenberg group under the non-proper action of CR automorphism group. 
 

In contrast with integrable case, the  transformation formula of the pseudo-Hermitian scalar curvatures is much more complecated than the CR Yamabe equation, if the CR structure is not integrable and this complexity makes it difficult to follow the analysis of integrable case.
  
In this talk, I will introduce a sub-class of contact sub-Riemannian manifolds for which sub-conformal transformation formula of a twisted sub-Riemannian scalar curvature becomes the sub-conformal Yamabe equation. Using the sub-conformal Yamabe equation, we will also discuss about the characterization 5 and 7-dimenisional non-integrable strongly pseudo-convex almost CR manifolds under the non-proper action of CR automorphisms by Schoen's argument.

The proof of existence can be done reliably by computers using interval arithmetics.We showed the verification of the bifurcated solutions of heat convection problems in 3D, and tried to show the existence of the band gap of photonic crystals.

이번 발표에서는 사영 공간에 매립된 사영 대수 다양체 (혹은 그에 대응하는 polynomial ring의 homogeneous prime ideal)에 대한 차수(Degree), 방정식의 개수 그리고 Green-Lazarsfeld Index라는 불변량들이 만족하는 부등식을 설명하고자 합니다. 특히 부등식들에 등장하는 경계치에 가까운 다양체들의 분류에 대해서 최근까지의 결과들에 대해서 발표할 예정입니다.  

최근 우리 사회의 화두 중 하나는 '통섭'이다. 우리를 가두고 있었던 울타리를 넘나들며 새로운 이웃과 만나 서로가 가진 생각을 섞는다는 의미에서 통섭 현상은 바람직하며, 대학 구성원들이 진지하게 생각해 볼 가치가 있다. 그러나 통섭의 시대일수록 "좋은 담은 좋은 이웃을 만든다"는 프로스트의 지적이 의미를 가진다. 담은 너무 높아서도 안 되지만, 너무 낮아도 안 되기 때문이다. 나는 오래 전부터 '하이브리드'(hybrid)가 '순종'만을 강조하는 우리 사회에 꼭 필요한 세계관이자 철학이라고 강조했다. 그런데 하이브리드에 대한 오해는 불식되지 않는 듯 하다. 하이브리드는 담을 완전히 헐자는 얘기 아닌가? 하이브리드는 지저분한 '짬뽕'아닌가? 하이브리드에도 윤리 의식이 있는가? 본 강연은 이러한 질문에 답하면서, 우리 사회에 필요한 하이브리드의 가치를 다시 한 번 강조할 것이다.

The zeta-function of a variety defined over the integers encoded the number of solutions with coefficients in all finite fields. Surprisingly, the value of this zeta function is related to other interesting invariants of the variety. A good example is the analytic class number formula, which related the value of the Dedekind zeta-function of a number field to the class number and other invariants. We will discuss generalizations of this to varieties over finite fields.

Symplectic manifold is an even dimensional smooth manifold with a special 2-form (called a symplectic form)  which is nowhere vanishing and closed. For a given symplectic manifold (M, w) and a smooth function f : M -> R, there is a dual vector field of df with respect to w. (we call it "Hamiltonian vector field of f").  In many cases, a closure of the integral curve of Hamiltonian vector field is isomorphic to a connected abelian Lie group, i.e. a torus.

In this talk, we focus on the case when the closure of the orbit is a circle, and we discuss what topological data make the given circle action to be Hamiltonian.

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