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.

.

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

For the past several years, studies on affine processes have been worked out by many researchers about moment explosions, implied volatilities, and long-term behaviors. Recently, Glasserman and Kim, and Keller-Ressel investigated the moment explosions of the canonical affine models of Dai and Singleton, and general two-factor affine stochastic volatility models, respectively, and presented their long-term behaviors. On the other hand, Benaim and Friz, and Lee showed that implied volatilities at extreme strikes are linked to the moment explosions of stock prices at given option maturities. In this work, we characterize the regions in which moment explosions happen for some time or at a given time, and relate them to the long-term behavior of stock prices and to implied volatilities, extending previous works on moment explosions for affine processes.

Contact geometry can be regarded as an odd-dimensional counterpart of
symplectic geometry. In 3-dimensional case we have a dichotomy of
contact structures: tight vs. overtwisted. Overtwisted contact
structures are not very interesting from a topological point of view,
but tight structures have intimate and subtle relationship with
topology. In this talk, I introduce contact structures of odd
dimensional manifolds and some interesting results about contact
topology including the result of Colin, Giroux, and Honda that states a
3-manifold admits infinitely many tight contact structures if and only
if it has an incompressible torus.

Given a hypergraph H = (V,E), a coloring of its vertices is said to be conflict-free if for every hyperedge S ∈ E there is at least one vertex whose color is distinct from the colors of all other vertices in S. The study of this notion is motivated by frequency assignment problems in wireless networks. We introduce and study the list-coloring (or choice) version of this notion. Joint work with Panagiotis Cheilaris.

We'll discuss joint work with J. Colliander, G. Staffilani, H. Takaoka, and T.Tao. The particular result is motivated by the much harder goal of understanding how energy can be exchanged between the different modes (or `frequencies') of nonlinear partial differential equations (PDE). The energy of solutions to such PDE is often conserved, but one can ask whether it's possible for smoother norms to grow in time. Such growth provides at least some quantitative measure of how energy might cascade from lower modes to arbitrarily high modes of the solution - a phenomenon sometimes described as "turbulence".

Given a hypergraph H = (V,E), a coloring of its vertices is said to be conflict-free if for every hyperedge S ∈ E there is at least one vertex whose color is distinct from the colors of all other vertices in S. When all hyperedges in H have cardinality two (i.e., when H is a simple graph), this coloring coincides with the classical graph coloring. The study of this coloring is motivated by frequency assignment problems in wireless networks and several other applications. We will survey this notion and introduce some fascinating open problems.

Given a Hamiltonian Lie group action on a symplectic manifold, I will explain the construction of a quantum deformation of the Kirwan map. It is defined by counting solutions of the symplectic vortex equations, which are a gauge theoretical deformation of the Cauchy-Riemann equations. Based on this map, I will formulate a quantum abelianization conjecture (joint work with Chris Woodward).

Recent work of the authors and colleagues on the turbulent mixing of compressible fluids is developed
and extended with an emphasis on the multiscale aspects of this work.

Hilbert proposed his 12-th problem at the Paris ICM in 1900 that

"it may be possible to find, for an arbitrary number field $K$, a transcendental function whose values generate the abelian extensions of $K$."

 In 1920's Takagi established the existence of Class Fields. Given a number field $K$ and a generalized ideal class group $G$ of $K$, his theory asserts that there exists a unique abelian extension of $K$ with $G$ its Galois group. Such an abelian extension is called the

class field of $K$ corresponding to $G$. Unfortunately, his argument could not provide us any explicit algorithm to construct class fields.

Therefore, we shall talk about Complex Multiplication Theorem which enables us to construct class fields over imaginary quadratic fields explicitly, due to Kronecker, Hecke, Hasse, Deuring and Shimura.

There is no general theory for solving nonlinear partial differential equations. The calculus of variations and maximum principle are useful methods to solve such nonlinear elliptic partial differential equations.
In this talk, I will briefly review the basic concepts of the calculus of variations and maximum principle and prove the nonexistence of solutions for some elliptic system using the maximum principle.

In this talk, we discuss different approaches to the
well-posedness problem of the one-dimensional cubic nonlinear
Schrodinger equation (NLS.)  First, we review how one can obtain local
well-posedness in H^s, s>1/2, by energy method and in L^2 by
Bourgain's periodic Strichartz estimate.  Then, we show that NLS is
"ill-posed" below L^2.

In order to construct solutions below L^2, we renormalize the
nonlinearity (called Wick ordering.)   Then, we consider the
renomarlized NLS with randomized initial data and show that it is
locally well-posed almost surely in H^s, s> -1/3.  Finally, we discuss
how to extend such solutions to global ones by combining probabilistic
local well-posedness theory with Bourgain's high-low argument (= an
argument for showing (deterministic)  global well-posedness, developed
by Bourgain, and it is a precursor to the I-method.)  This part is a
joint work with James Colliander (University of Toronto.)

There is a summary of these results in a survey paper with Catherine
Sulem (University of Toronto.)

We confirm that Entropy Conjecture holds
for every continuous self-map
of a compact K( π, 1) manifold
with the fund. gr. 

π virtually nilpotent,
e.g. for every continuous map of an infra-nilmanifold.
In fact, a lower estimate of the topological entropy
by the logarithm of the spectral radius of exterior power
of an assoc. ”linearization matrix” with integer entries.
This & estimates of Mahler measure of polynom.,
give some absolute lower bounds for the entropy.

Manin showed how the Brauer group of schemes together with class field theory(reciprocity laws) accounts for many counterexamples to the Hasse principle(local-global principle) and to weak approximation for rational points of projective varieties. Only recently did one start to investigate  an analogous approach for integral points of affine varieties. Here the relevant approximation property is strong approximation,a generalization of the chinese remainder theorem. I shall recall what is known or conjectured for rational points.
I will then go on to describe what has been achieved for integral points :
unconditional results for many classes of homogeneous spaces of linear algebraic groups,
with a connexion to the classical study of integral quadratic forms(F. Xu and the speaker, D. Harari, M. Borovoi, C. Demarche) ; computations and conjectures for curves (D. Harari and F. Voloch) ; computations for certain  cubic surfaces (O. Wittenberg and the speaker).
I will in particular discuss the classical problem of representation of an integer as sum of three cubes of integers.

In this talk, we consider the relativistic Vlasov equations coupled with some nonlinear electromagnetic fields. The nonlinear electromagnetic fields considered here can be regarded as some special cases of the nonlinear electromagnetic field of the Born-Infeld theory. We will show that classical solutions exist globally in time for some
restricted cases.

In this seminar, I will introduce to wave equation. Wave equation is an important partial differential equation that describes the propagation of a variety of wave, such as sound waves, light waves and water waves.

Usually, I proved the existence of solutions of wave equation by using Faedo-Galerkin method. I shall explain to Faedo-Galerkin method. Next, I will apply to wave equation by using Faedo-Galerkin method.

The Boltzmann-BGK model is widely used in the kinetic theory of gases as a qualitatively correct model for the Boltzmann equation. Recently, a semi-Lagrangian scheme for the BGK model was suggested and tested successfully for various flow problems arsing in gase dynamics.

In this talk, I will breifly review the basic aspects of collisional kinetic equations and show that the approximate distribution function of the scheme converges to the smooth solution of the BGK model.

Solitons are nonlinear solutions that maintain its shape and travel at a constant speed. They widely arise in focusing nonlinear dispersive equations. They are occurred as a balance of nonlinear reinforcing effects and dispersive effects in the medium. In this talk, I will briefly review long-standing history of solitons and discuss past and current issues around soliton solutions. It will include a brief explanation to the inverse scattering method, the orbital (asymptotic) stability, and then an introduction to the 'soliton resolution conjecture'.

In a non-relativistic case, the dynamics of N-particle system is governed by the Schrödinger equation. The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this talk, I will introduce estimates on the rate of convergence of the quantum mechanical evolution towards the Hartree dynamics. Some basic concepts and tools of mathematical physics will also be covered.