Many problems that are intractable on general graphs become linear time solvable on graphs of bounded treewidth. The constant factor however of such algorithms is exponential or worse in the treewidth of the tree decomposition that is used. In this talk,  a number of known and some new results are surveyed. In particular, it is shown how speed improvements can be obtained using convolutions, and how a recent technique called "cut-and-count" can be used to obtain fast probabilistic algorithms for problems like Hamiltonian Circuit.

In this talk, we investigate the structure of highest weight
modules over the quantum queer superalgebra $U_q(q(n))$ and develop
the crystal basis theory for $U_q(q(n))$. We define the notion of
crystal bases and prove the tensor product rule for
$U_q(q(n))$-modules in the category $O_{int}^{\ge 0}$. Moreover, we
give an explicit combinatorial realization of the crystal $B(\lambda)$
for an irreducible highest weight $U_q(q(n))$-module $V(\lambda)$ in
terms of semistandard decomposition tableaux.

The Rogers-Ramanujan identities have many natural and
significant generalizations. The generalization we will present in
this talk was first studied by D. Bressoud, by considering the
partitions he named as ”footed partition”. Bressound then made a
conjecture that two sets of partitions under certain constraints are
equinumerous. The validity of the conjecture in the first two cases implies
exactly the partition-theoretical interpretation for the Rogers-Ramanujan
identities. We give a nearly bijective proof of the conjecture, and we
provide examples to demonstrate the bijection as well.

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

In order to understand complex geometry, it can be useful
to construct "pathological" examples. Nonkähler or nonprojective
compact complexe manifolds are such examples. Hopf (in 1948) and
Calabi and Eckmann (1953) constructed complex structures on product of
spheres with odd dimension which are not kähler. In this talk, we will
present a generalization due to Bosio of these manifolds. These
manifolds are called LVMB manifolds are their interest lies in the
very combinatorial nature. In particular, we will show that LVMB
manifolds are homeomorphic to quotients of important objects in toric
topology, namely moment-angle complexes (MAC).We will also show the
relationship between LVMB manifolds and toric (algebraic) varieties.

A posteriori error estimation produces qualitative and/or quantitative information on the numerical errors and now plays a fundamental role in adaptive mesh refinement for efficient implementation of numerical methods. Recently, there has been growing interest in the type of a posteriori error estimation which provides a fully computable upper bound on the acutal error. In this talk we will give an overview of some recent results on this issue for primal and mixed finite element methods.

This is an elementary introduction to Langlands program, focusing
on simple examples of GL(2) and modular curves. We will start with
the definition of modular forms and explain their relation to automorphic
representations of GL(2). After defining L-groups and automorphic
L-functions, we will state the Langlands' principle of functoriality and
explain for the cases of Jacquet-Langlands correspondence and
base change. One of main tools used to prove these cases of functoriality
is the trace formula. We will explain this trace formula (or rather simplified
version of it) and another application of it to show that the Hasse-Weil zeta
function of modular curves can be expressed as a product of automorphic
L-functions.

This is an elementary introduction to Langlands program, focusing
on simple examples of GL(2) and modular curves. We will start with
the definition of modular forms and explain their relation to automorphic
representations of GL(2). After defining L-groups and automorphic
L-functions, we will state the Langlands' principle of functoriality and
explain for the cases of Jacquet-Langlands correspondence and
base change. One of main tools used to prove these cases of functoriality
is the trace formula. We will explain this trace formula (or rather simplified
version of it) and another application of it to show that the Hasse-Weil zeta
function of modular curves can be expressed as a product of automorphic
L-functions.

We generalize two bijections due to Garsia and Gessel to compute the generating functions of the two vector statistics (des_G, maj, ℓ_G, col) and (des_G, ides_G, maj, imaj, col, icol) over the wreath product of a symmetric group by a cyclic group. Here des_G, ℓ_G, maj, col, ides_G, imaj_G, and icol denote the number of descents, length, major index, color weight, inverse descents, inverse major index, and inverse color weight, respectively. Our main formulas generalize and unify several known identities due to Brenti, Carlitz, Chow-Gessel, Garsia-Gessel, and Reiner on various distributions of statistics over Coxeter groups of type A and B.

In order to understand complex geometry, it can be useful
to construct "pathological" examples. Nonkähler or nonprojective
compact complexe manifolds are such examples. Hopf (in 1948) and
Calabi and Eckmann (1953) constructed complex structures on product of
spheres with odd dimension which are not kähler. In this talk, we will
present a generalization due to Bosio of these manifolds. These
manifolds are called LVMB manifolds are their interest lies in the
very combinatorial nature. In particular, we will show that LVMB
manifolds are homeomorphic to quotients of important objects in toric
topology, namely moment-angle complexes (MAC).We will also show the
relationship between LVMB manifolds and toric (algebraic) varieties.

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

In 1979, Paul Yang showed that the bidisc does not admit any complete K ̈ahler metric with its bisectional curvature bounded between any two negative constants. His proof works for any dimensional polydiscs as well as the Hermitian symmetric domains with rank not less than 2. This method works also for product of complete K ̈ahler manifols, as shown by H. Seshadri and F.Zheng. Still, whether there are examples falling into this category that are neither homogeneous nor product is poorly understood. In this talk, I will present new examples of inhomogeneous bounded domains that cannot admit complete Kahler metrics with their bisectional curvature bounded between any prescribed negative constants.

Transportation and network polytopes are classical objects in operations research. In this talk, we focus on recent advances on the diameters of several classes of transportation polytopes, motivated by the efficiency of the simplex algorithm. In particular, we discuss results on 2-way transportation polytopes, including a recent result of Stougie and report on joint work with Bruhn-Fujimoto and Pilaud, concerning 2-way transportation polytopes with a certain support structure. We also present a bound on 3-way transportation polytopes in joint work with De Loera, Onn, and Santos. To conclude, we discuss avenues for future work on transportation polytopes and their diameters.

We survey our recent results on rigidity vs flexibility of representations in semisimple Lie groups and group actions on character variety.

인터넷의 가장 중요한 문제중 하나인 혼잡제어 문제를 바라보는 수학적인 시각에 대해서 얘기합니다. 협조(cooperation)와 경쟁(competition)의 논리에서 어떻게 인터넷을 제어할 수 있는가에 따라서, 바라보는 문제의 다름에 대해서 논하고, 최적화 이론과 게임 이론과 같은 수학적 도구가 어떻게 인터넷을 발전시키는데 공헌을 하는지에 대해서 얘기합니다.

※5시부터 시작전 다과회(피자제공)가 있습니다.

In coherent imaging systems, such as synthetic aperture radar
(SAR), the observed images are contaminated by multiplicative noise. Due
to edge preserving feature of the total variation (TV), variational
models with TV regularization have attracted much interest in removing
multiplicative noise. However, the fidelity term of the variational
model, based on maximum a posteriori estimation, is not convex and so it
is usually difficult to find a global solution. In this talk, we
introduce how to relax the nonconvexity of the variational model.
Algorithm based on the augmented Lagrangian function can be applied to
solve our proposed model. But this algorithm requires to solve a
subproblem, which does not have closed form solution, at each iteration.
Hence we adapt the linearized proximal alternating minimization
algorithm (LPAMA) which does not require any inner iterations. Also, the
proposed method is very simple and highly parallelizable and thus it is
efficient to remove multiplicative noise in huge SAR images.
models with TV regularization have attracted much interest in removing
multiplicative noise. However, the fidelity term of the variational
model, based on maximum a posteriori estimation, is not convex and so it
is usually difficult to find a global solution. In this talk, we
introduce how to relax the nonconvexity of the variational model.
Algorithm based on the augmented Lagrangian function can be applied to
solve our proposed model. But this algorithm requires to solve a
subproblem, which does not have closed form solution, at each iteration.
Hence we adapt the linearized proximal alternating minimization
algorithm (LPAMA) which does not require any inner iterations. Also, the
proposed method is very simple and highly parallelizable and thus it is
efficient to remove multiplicative noise in huge SAR images.

Toric varieties are in one-to-one correspondence with rational fans.
This establishes a useful bridge between algebraic geometry and convex
geometry. Algebraic geometers can translate many problems into
statements which are more amenable to computations. Conversely, and
even more remarkably, some combinatorial questions can be successfully
reformulated in algebraic-geometric terms. I will describe a model of
toric varieties associated to nonrational fans, with supporting
evidence that the rich interplay between algebraic and convex
geometries carries over. This is a joint work with F. Battaglia.

Groups, Graphs and Geometry

- a gentle introduction to circle packing

평면이나 구면 상에 겹치지 않는 원판들이 있을 때, 그 접점 관계(tangency relation)를 나타내는 그래프는 “동전 그래프”라 불립니다. 이 강연에서 우리는 Koebe (1936) – Andreev (1970) – Thurston (1978)에 의해 밝혀진 다음 정리의 증명을 살펴 봅니다.
정리. 임의의 평면 그래프는 동전 그래프이다.
학부 수준의 복소 해석학만으로 증명할 수 있는 이 아름다운 정리는 다면체에 맞는 원형 철망을 씌우는 문제, 다각형의 예각 삼각형화를 찾는 문제, 3차원 다양체를 기하화하는 문제 등과 연관됩니다. 이는 기하학적 군론, 그래프 이론, 위상수학을 어우르는 주제입니다.

 Text : Basic Analytic Number Theory 

        by A. Karatsuba

If one draws the Hasse diagram for the face lattice of a polytope, this may appear to be the 1-skeleton of some other polytope. In 1971 Lindström asked whether the intervals of a polytope’s face lattice ordered by containment can be realized as the face lattice of another polytope. If so, this would be the polytope appearing the Hasse diagram. In this talk, I will give necessary and sufficient conditions for the intervals of a polytope’s face lattice to be realizable, and I will provide a counter example showing that not every polytope satisfies these conditions.

 복수측정벡터 문제는 0이 아닌 성분이 공통된 위치에 분포하는 성긴 벡터들을 복원하는 방법을 다루는 문제이며, 도착방향 추정문제, 산란광단층촬영, 역산란 문제 등 여러 실용적인 문제에 응용될 수 있다. 2000년대 중반부터 신호처리와 정보이론 분야 등에서 주목을 받기 시작한 압축감지 기법은, 성긴 속성을 갖는 벡터들을 샤논 샘플링 비율보다 상당히 낮은 샘플링 비율로도 높은 확률로 정확히 복원할 수 있다는 점을 보여주고 있으며, 이로 인해 복수측정벡터 문제에 압축감지 기법을 응용하는 일련의 연구들이 이어져 왔다.
본 세미나에서는 먼저 압축감지 기법의 개념과 주요한 결과들을 소개하고, 복수측정벡터 문제에 대한 압축감지 기법의 연구결과에 대해 살피도록 한다. 또한, 압축감지 기법을 이용한 복수측정벡터 문제에 대한 알고리즘은 측정벡터의 개수가 늘어날수록 성능향상이 둔화되어, 일정 갯수 이상의 측정벡터에 대해서는 배열신호처리 기반의 복수신호분리(MUSIC) 알고리즘보다 성능이 더 나쁜 것을 볼 수 있는데, 이러한 문제점을 해결하기 위해 연구한 최근의 결과에 대해서도 살펴보고자 하며, 이는 기존의 압축감지 알고리즘 혹은 배열신호처리 알고리즘을 개별적으로 복수측정벡터 문제에 적용할 때 생기는 한계를 극복할 수 있게 해 준다.

In this talk, we will explain a Hitchin system in an elementary way. That is, we will review basics in symplectic geometry such as a moment map, the Marsden-Weinstein quotient, etc in a finite dimensional setting. After reviewing these materials and related elementary examples, we will sketch out how theses can be generalized
to the comparable ones in an infinite dimensional setting, i.e., in a gauge theoretic way.

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

The phase transition deals with sudden global changes and is observed in many fundamental problems from statistical physics, mathematics and theoretical computer science, for example, Potts models, graph colourings and random k-SAT. The phase transition in random graphs refers to a phenomenon that there is a critical edge density, to which adding a small amount a drastic change in the size of the largest component occurs. In Erdös-Renyi random graph, which begins with an empty graph on n vertices and edges are added randomly one at a time to a graph, a phase transition takes place when the number of edges reaches n/2 and a giant component emerges. Since this seminal work of Erdös and Renyi, various random graph models have been introduced and studied. In this talk we discuss phase transitions in several random graph models, including Erdös-Renyi random graph, random graphs with a given degree sequence, random graph processes and random planar graphs.

지금은 융합과학기술의 시대라고 일컫는다. 하지만 아무도 융합이 무엇을 연구하는 분야인지그 정의를 명확하게 내리지는 못한다. 그 이유는 아마도 융합은 학문이나 기술의 분야가 아니라 그들을 하고자하는 방법론이어서 그럴 것이다. 수학을 바탕으로 자연과학이 나타났고 또 이들을 바탕으로 공학분야가 탄생하였다. 수 많은 학문이나 기술 분야가 각각 많은 발전을 하였고 산업에 이바지하였다. 이제 산업이 고도화되고 사회의 요구성이 매우 다양해져서 이러한 니즈를 맞추기에 단독적인 학문이나 기술로는 턱없이 부족하기에 이르렀다. 학문적으로도 새로운 생각이나 방법이 단독 분야에서 나타나는 것이 아니라 여러 분야의 화학적인 결합으로 태어난다. 따라서 어쩌면 융합이라는 분야는 무엇이 될지 모르는 정의될 수 없는 것이 당연할지 모른다.

현재 비교적 융합이 나타나는 분야는 우리의 가장 큰 관심사인 건강한 삶을 주제로 한 분야이다. 암 치료를 위한 첨단 의료기기로부터 비만 관리에 이르는 ‘라이프사이언스’에 이제까지의 모든 학문적 기술적 지식이 요구된다고 할 수 있다. 본 콜로퀴엄에서는 필자가 수행하였던 ‘라이프사이언스’ 연구 관련 주제를 소개하고 이들의 방향을 토의해보고자 한다. 여기에는 유전자나 단백질 등 유전정보의 기능을 예측하거나 분석하고 관리해주는 바이오인포매틱스, 정보통신 기술을 바탕으로 건강 관리를 해주는 u-헬스케어, 의사에게 객관적인 정보를 주어 진단의 정확도를 높일 수 있는 컴퓨터도움진단, 그리고 양성자치료기와 같은 첨단 의료기기 등의 분야가 포함된다. 덧붙여서 필자가 근무하였던 한국연구재단에서 보아왔던 연구자들의 융합분야도 소개하고자 한다.

수학은 모든 학문과 방법론의 기본이 된다. 수학적 지식과 마음가짐으로 타 분야와의 융합을 통하여 이제까지 볼 수 없었던 새로운 학문의 지평선을 열기를 기대해본다. 본 콜로퀴엄은 이러한 취지에서 융합과학기술의 일부분의 예를 소개하는 것이고 더욱 창의적인 융합의 지평선을 카이스트에서 열어주길 기대한다.

In the first half of this talk, we will review the notion of affine processes on positive semidefinite matrices, which were introduced by Cuchiero et al (2011). These processes have been applied to various modeling problems in the finance literatures. In the latter half, I will introduce our recent research result on affine processes. We establish the relationship between the Laplace transforms for linear functionals of affine processes on positive semidefinite matrices and the generalized Riccati integral equations. The result generalizes and unifies the affine transform formula and the systematic exploitation of Laplace transforms of squared Bessel functionals in a natural way. As consequences of the result, we found some formulae such as Laplace transforms of the marginal distributions, transition densities, and Pitman-Yor's formula for Wishart processes. Moreover, we also present the conditional Laplace transforms of Wishart functionals.

In one of their joint papers, Victor Reiner and Dennis Stanton introduced a (q,t)-generalization of the binomial coefficient. There was an interesting conjecture for the cases when q≤-2 is a negative integer. In this talk, I will prove this conjecture and try to give some combinatorial sense using integer partitions.

국내외 금융시장에서 거래되고 있는 파생상품은 종류가 매우 다양해지고 구조가 복잡해지고 있는 추세이며, 이에 따라 새로운 수학적 모델과 기법의 도입이 필요하다.

본세미나에서는 파생상품에 대한 기존의 수학적 모델과 가격 결정이론에 대해 설명하고, Fourier 변환을 이용한 Semi-closed form 형태의 파생상품 가격결정이론을 소개한다.

학부에서 왜 현대대수학 I, II를 온전히 배워야 하는지를, BC 5세기경 Pythagoras로부터 시작된 수론문제가 현재까지 어떻게 발전되어 왔는지를 예로 들어 설명합니다.

In this lecture series, we will cover basic notions of a Hitchin
system and related topics with some variations and cadenzas. . The
lectures will be generically in a expository nature: The main article
to study in the lecture series is N. Hitchin’s paper “Stable bundle
and Integrable systems”. If time permits, we will also give some
remarks about other papers “Lie groups and Teichmuller space” and
“Langlands duality and $G_2$ spectral curves”.

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

Many combinatorial problems and arguments concerning finite point sets in the Euclidean plane (or higher dimensions) often do not use the linear structure. A more general concept is that of an Oriented Matroid (OM). It is well-known that every OM can realized by pseudolines, and in fact most oriented matroids can not be realized by straight lines.
Recently, Alfreod Hubard (Courant Institute) and myself have found a new way to represent an OM by convex sets which retains much more of the “straightness” of the Euclidean plane. Interestingly, in our model the isotopy conjecture holds in a very strong sense, and it unifies several aspects of pseudoline arrangements.

 Deligne observed a long time ago that the category of split real Hodge structures is equivalent to the category of representations of a certain rank 2 torus.  Later he described the Tanakian Galois group of the category of all real mixed Hodge structures as a certain extension of this torus.  I'll discuss this and joint work with Greg Pearlstein where we spell out a similar picture for degenerations of real mixed Hodge structure.

There's something about a symmetric shape that really grabs our attention, particularly if you're a mathematician. In this talk I will discuss what a mathematician sees when they look at a symmetric object, how they abstract out the essence of the situation and what one could possibly mean by the statement that "there are only 17 types of wallpaper." Applications of these abstract structures to physics, chemistry and computer science will also be discussed.

대칭적인 모양에는 사람들의 시선을 끄는 특별한 힘이 있습니다. 이 강연에서는 수학자들이 대칭적인 물체를 바라보는 방법에 대해 얘기해 보고자 합니다.

보통 사람들이 느끼는 대칭성을 수학자들이 어떻게 추상화하고, 그 추상화의 능력을 통해 이 세상의 모든 벽지를 오직 17개의 종류 만으로 분류할 수 있었는지 소개합니다. 이러한 추상적인 구조는 물리학, 화학, 전산학 등에서도 이용됩니다.

We propose a computational framework to incorporate regularization terms used in regularity based variational methods into least squares based methods. In the regularity based variational approach, the resulted image is obtained as a result of the competition between the fidelity term and a certain regularity term, while in the least squares based approach the image is computed as a minimizer to a constrained least squares problem.

The total variation minimizing denoising scheme is an exemplary scheme of the former approach with the total variation minimizing term as the regularity term, while the moving least square method is an exemplary scheme of the latter approach.

Both approaches have appeared in the literature of image processing independently.

By putting schemes from both approaches into a single framework, the resulted scheme benefits from the advantageous properties of both parties. As an example, in this talk, we propose a new denoising scheme, where the total variation minimizing term is adopted by the moving least squares method. The resulted denoising scheme overcomes the drawbacks of both schemes, i.e., the staircasing artifact in the total variation minimizing based denoising and the local boiling artifact in the moving least square based denoising method.

The proposed computational framework can be utilized to put various combinations of both approaches with different properties together.

I will discuss the existence of injective maps between right-angled Artin groups.  In the case of two-dimensional right-angled Artin groups, I will give a complete classification of right-angled Artin subgroups of right-angled Artin groups.  I will then derive some new rigidity results.  This is joint work with Sang-hyun Kim.

I will talk about a degenerate diffusion Patlak-Keller-Segel system in $n\geq 3$ dimension. The main difference between the current work and many recent works on the same model is that we study the diffusion exponent $m=2n/(n+2)$ which is smaller than the exponent $m^{*}=2-2/n$ used in those recent works. With the exponent $m=2n/(n+2)$,  the associated free energy is conformal invariant and there is a family of stationary solution $U_{\lambda,x_0}(|x|) =C(n)(\frac{\lambda}{\lambda^2+|x-x_0|^2})^{\frac{n+2}{2}}$, $\forall \lambda>0$. For radially symmetric solutions, we prove that if the initial data is strictly below $U_{\lambda,0},(|x|)$ for some $\lambda$ then  the solution vanishes in $L^1_{loc}$ as $t\to\infty$; if the initial data is strictly above $U_{\lambda,0} (|x|)$ for some $\lambda$ then  the solution concentrates at $r=0$  as $t\to\infty$. We then prove that there is a global weak solution provided that the $L^m$ norm of initial density is less than a universal constant, and the weak solution vanishes as time goes to infinity. We also prove a finite time blow up of the solution if the $L^m$ norm for  initial data is large then that of $U_\lambda(|x|)$ and the free energy of initial data smaller than that of $U_\lambda(|x|)$.