For a positive integer $N$ divisible by $4,5,6,7$ or $9$, let $mathcal{O}_{1,N}(mathbb{Q})$ be the ring of weakly holomorphic modular functions for the congruence subgroup $Gamma_1(N)$ with rational Fourier coefficients.

 We present explicit generators of the ring $mathcal{O}_{1,N}(mathbb{Q})$ over $mathbb{Q}$ by making use of modular units which have infinite product expansions.

  A family of algebraic curves covering a projective variety X is called a web of curves on X if it has only finitely many members through a general point of X. A web of curves on X induces a web-structure, in the sense of local differential geometry, in a neighborhood of a general point of X.  We willdiscuss the relation between the local differential geometry of the web-structure and the global algebraic geometry of X.

(1)Symplectic vector spaces

(2) Symplectic manifolds

(4) Symplectomorphism groups

(5) Hamiltonian diffeomorphisms

(6) Moment maps

(7) Equivariant cohomology

(8) Duistermaat-Heckman theorem

(9) Open problems

동역학계의 질문들은 천체의 움직임에서 출발했으나, 원을 돌리는것이나 당구공과 같은 아주 쉬운 예들에서도 많이 있다.  그러나 그들에 대한 성질들은 최근에 와서야 이해되기 시작하였다.  동역학계의 발전과정, 그리고 예들을 통하여 여러 종류의 섞임(mixing)에 관한 정의와 성질들을  살펴본다.

참석하고자 하시는 분은 아래 링크를 통해 사전등록을 해주시면 감사하겠습니다^^

http://goo.gl/00Ev5r

(1)Symplectic vector spaces

(2) Symplectic manifolds

(4) Symplectomorphism groups

(5) Hamiltonian diffeomorphisms

(6) Moment maps

(7) Equivariant cohomology

(8) Duistermaat-Heckman theorem

(9) Open problems

삼성전자 Software 분야에 대한 소개 및 현재 연구/개발에서 사용되고 있는 수학분야에 대해 소개한다. 최근 인공지능 분야가 학계 및 업계에서 각광을 받고 있는 이유와 기계학습(Machine Learning) 분야에 대해 소개한다.

This is a followup talk tomy CS colloquium on March 2. In that talk I discussed the problems of counting independent sets and colorings. Those problems are examples of antiferromagnetic systems in which neighboring vertices prefer different assignments. In this talk we will look at ferromagnetic systems where neighboring vertices prefer the same assignment. We will focus on the ferromagnetic Potts model. We will look at the phase transitions in this model, and their connections to the complexity of associated counting/sampling problems and the performance of related Markov chains.

(1)Symplectic vector spaces

(2) Symplectic manifolds

(4) Symplectomorphism groups

(5) Hamiltonian diffeomorphisms

(6) Moment maps

(7) Equivariant cohomology

(8) Duistermaat-Heckman theorem

(9) Open problems

대학원생을 위한 영역분할법 특강 I

Domain decomposition methods are iterative methods for solving the often very large linear or nonlinear systems of algebraic equations that arise in various problems in mathematics, computational science, engineering and industry. In this talk, we would like to give an overview over overlapping Schwarz methods that are based on partitioning the domain of the problem into over-lapping subdomains, solving local problems on these subdomains, and solving an additional coarse problem associated with the subdomain mesh [1, 2].

(1)Symplectic vector spaces

(2) Symplectic manifolds

(4) Symplectomorphism groups

(5) Hamiltonian diffeomorphisms

(6) Moment maps

(7) Equivariant cohomology

(8) Duistermaat-Heckman theorem

(9) Open problems

Let $n$ be an any integer. It is well known that there are infinitely many imaginary quadratic fields with ideal class group having a subgroup isomorphic to $Z/nZ times Z/nZ$. For real quadratic fields, less is known. We will prove that there exist infinitely many real quadratic number fields with ideal class group having a subgroup isomorphic to $Z_n times Z_n$. We will also prove that there exist infinitely many imaginary quadratic number fields with ideal class group having a subgroup isomorphic to $Z_n times Z_n times Z_n$.

대학원생을 위한 영역분할법 특강

 Domain decomposition (DD) methods are numerical methods for partial differential equations that provide a natural approach to solving large scale problems on parallel computers. Firstly, domain decomposition is in a form of divide-and-conquer for mathematical problems posed over a physical domain. In DD, a large problem is reduced to a collection of smaller problems, each of which is easier to solve computationally than the undecomposed problem, and most or all of which can be solved independently and concurrently. Typically, it is necessary to iterate over the collection of smaller problems, and much of the theoretical interest in DD methods lies in ensuring that the number of iterations required is very small. Secondly, DD is often a natural paradigm for the modeling community. Physical systems are decomposed into two or more contiguous subdomains based on phenomenological considerations and the subdomains are discretized accordingly, as independent tasks.

 SU(3) Chern-Simons theory in 2+1 dimension is a gauge theory with SU(3) gauge group. This theory exihibits several types of vortex solutions. Among others, we discuss vortex solutions of A-nontopological type, of which some component is topological and the other is topological. Developing variational type argument together with bubbling analysis, we give existence of such solutions.

Orthogonal polynomials are a family of polynomials which are orthogonal with respect to certain inner product. The n-th moment of orthogonal polynomials is an important quantity, which is given as an integral. In 1983 Viennot found a combinatorial expression for moments using lattice paths. In this talk we will compute the moments of several important orthogonal polynomials using Viennot's theory. We will also see their connections with continued fractions, matchings, set partitions, and permutations.

There has been work in either OM or Statistics addressing the problem of how call centers – and other high volume service businesses – can better manage the capacity-demand mismatch that results from arrival-rate uncertainty. OM papers account for uncertainty when making staffing and scheduling decisions. Statistical models have sought to better characterize the distribution of arrival rates, by time of day, as they evolve. While each line of research has made important progresses in addressing certain elements of the problem caused by arrival-rate uncertainty, neither addresses the whole problem. We present a data-driven integrated forecasting and stochastic programming framework to cope with arrival-rate uncertainty, from call centers with single arrival streams to those with multiple dependent arrival streams. Experiments with simulated and real call-center data highlight both operational and forecasting benefits of the integrated approach.

Around 15 years ago, Aharoni and Haxell gave a wonderful generalization of Hall's marriage theorem. Their proof introduced topological methods in matching theory which were further developed by Berger, Meshulam, and others. Recently, motivated by some geometric questions, we extended these methods further, and in this talk I'll explain the ideas and some of our results.

Smooth projective surfaces with rational Homology the same as the projective plane were objects of interest for a long time. In their seminal work Prasad and Yeung classified all such surfaces. When one allows singularities on such surfaces the problem becomes much harder. An interesting restriction (suggested by Kollar) is the existence of rational curves with only Cuspidal singularities in the smooth locus of the surface. We will discuss why the restriction is important and investigate the consequences it has for the kind of singularities the surface can admit.

  이 세상에 불확실성이 없으면 그것을 다룰 방법을 고민할 필요가 없을 것이다. 그러나 우리 주위에는 진로 선택할 때의 ‘불확실성’과 같은 현상이 매우 많다. 이 불확실성을 수학적으로 모형화하고 분석하는 방법을 다루는 학문이 통계학이다. 여기에 확률론은 바탕이 되는 이론을 제공해준다.

  우리는 기본적으로 일상 생활에서 통계를 한다. 불확실한 상황에 대해서 친구와 의논을 하거나 전문가와 상담을 받고 의사결정을 한다. 복잡한 상황을 단순화시키기도 하고, 상황의 핵심적인 요소를 찾으려고 노력한다. 요소가 여럿일 때는 이 요소들의 상호관계를 각자의 방식으로 정리한다. 이렇게 하고 나면 그 상황을 더 잘 이해할 수 있게 되고 관련된 미지의 상황에 대해서 예측이나 추론을 할 수도 있다. 이러한 작업들을 수학적 틀 안에서 전개할 수 있는 이론을 통계학에서 다룬다.

(1)Symplectic vector spaces

(2) Symplectic manifolds

(4) Symplectomorphism groups

(5) Hamiltonian diffeomorphisms

(6) Moment maps

(7) Equivariant cohomology

(8) Duistermaat-Heckman theorem

(9) Open problems

수학의 흥미 있는 점 중에 하나는 서로 다른 분야에서도 비슷한 내용의 결과가 성립한다는 것이다. 본 강연에서는 해석학적 정수론의 소수의 분포 정리(prime number theorem)에 대응하는 기하학적 결과인 prime geodesic theorem을 소개하고자 한다. 또한 리만제타 함수에 대응되는 기하학적인 제타 함수도 알아 본다.

참석하고자 하시는 분은 아래 링크를 통해 사전등록을 해주시면 감사하겠습니다^^
http://goo.gl/vhdz8V

3rd day (April 3rd, 3 hours):

Abstract: The Weil-Petersson form is a Kaehler form over the Teichmuller space of Riemann surfaces.In 1991, Takhtajan and Zograf discovered another Kaehler form over the Teichmuller space when the Riemann surface has punctures. In this talk, I will explain potential functions of these Kaehler two forms, which arethe Liouville action functional for Weil-Petersson form and a new pontential for Takhtajan-Zograf form.

This is a joint work with Takhtajan and Teo.

다변량 정규 확률벡터의 곱 적률은 통계학과 확률론에서 널리 필요한 통계량 가운데 하나이다. 다변량 정규 확률벡터의 차원이 n이면, 쉽게 알 수 있듯이, 곱 적률은 일반적으로 n차원 적분으로 나타낼 수 있으며, n이 1일 때와 2일 때에는 그 결과가 잘 알려져 있다. 또한, 그 밖의 경우 가운데에서 E{X_1 , X_2 , ... , X_n}은 1918년에 그 결과가 발표된 바 있다. 그 뒤, 한 세기에 걸쳐 여러 연구자들이 가장 일반적인 꼴인 E{ X_1^{a_1} , X_2^{a_2} , ... , X_n^{a_n} }을 얻고자 노력해 왔다. 이 발표에서는 한 세기 동안 여러 사람을 매혹시켜 온 이 문제를 다루어, 다변량 정규 확률벡터의 곱 적률의 닫힌 꼴을 얻어 본다.

For a finite graph Gamma, let G(Gamma) denote the right-angled Artin group on the complement graph of Gamma. In the talk, we introduce the notions of ``induced path lifting property'' and ``semi-induced path lifting property'' for coverings of graphs, and apply them to embedability between right-angled Artin groups.

  We recover the result of Sang-hyun Kim and Thomas Koberda that for any finite graph Gamma, G(Gamma) admits a quasi-isometric group embedding into G(T) for some tree T. The upper bound for the number of vertices of T is improved from a double exponential function in the number of vertices of Gamma to an exponential function.

2nd day (April 1st, 1.5 hours):

  A naive definition of mathematics may be “the science of quantity and space”. And in the simpler explanation, they correspond to arithmetic and geometry. One of modern dictionaries, Encyclopedia Britannica, defines it as “the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects” But it is more difficult to understand. Probably it may not be easy to define unanimously the mathematics. But we all know that the mathematics is an indispensable tool for the science and the engineering in modern times.
  The word engineering originated from the ingenium, may be defined as “the application of scientific, economic, social, and practical knowledge in order to invent, design, build, maintain, research, and improve structures, machines, devices, systems, materials and processes”. The engineering discipline is also extremely broad as the mathematics is. Therefore, the engineering area based on the classical mechanics will be dealt in this lecture. Since the early stage of mechanics was not much different from the physics and mathematics, the classical mechanics based areas like civil engineering, mechanical engineering, aerospace engineering, and etc., are now utilizing the mathematics most.
  In this lecture, the intersection and the differences of three disciplines, mathematics, science and engineering, will be explained after defining the three disciplines. And then some famous and important contributors to the classical mechanics, who were then mathematician and/or physicist, will be introduced and their connection with mathematics will be illustrated along with some equations. And then, the cooperation and the convergence of mathematics and engineering will be emphasized by introducing some representative examples.

We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we can define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a kind of tangle replacement. This method provides a way to classify Legendrian singular links.

2015 년 3 월 13 일(금) 오후 2시 00분 - 6시 00분

제목 : 이자율 파생 상품의 이해

강연자 : 전인태 (카톨릭대학)

장 소 : E6-1 #1409

초록 :

보통 금융기관은 천문학적인 금액을 이자율 관련 상품에 투자하고 있기 때문에 이자율 변동위험에 크게 노출되어 있다. 따라서 채권의 거래로부터 얻어지는 이자율의 구조 및 헤징 방법론은 금융기관의 최대의 관심사 중의 하나이고 금융공학에서 가장 중요하게 다루어지고 있다. 본 강의에서는 이자율 결정의 메커니즘, 이자율위험을 헤지하기 위한 이자율파생상품의 종류 및 구조, 가격결정 방법론 등에 대하여 소개한다.

This is a talk on T. Saito’s recent work.

For a sheaf on a surface, Deligne and Laumon defined the characteristic cycle implicitly using the total dimension of the nearby cycles under the “non-fierce” assumption. In general, T. Saito defined the characteristic cycle of a sheaf on a surface by combining the approach using the Radon transform and the non-logarithmic version of ramification theory developed by Abbes and Saito. In this talk, under the stability of nearby cycles and the existence of singular supports on a surface, we show how to construct characteristic cycle using Radon transform

Abstact: In this talk, we mainly introduce our recent work on the weight distributions of a class of cyclic codes from two distinct finite fields. We will show that these codes have three or four nonzero weights. Moreover, some optimal or nearly optimal cyclic codes are found.

An Abelian differential defines a flat structure on the underlying Riemann surface, such that it can be realized as a plane polygon with edges suitably identified. Varying the shape of the polygon induces a GL(2,R)-action on the moduli space of Abelian differentials, which is called Teichmueller dynamics. In this lecture series, I will give an elementary introduction to Teichmueller dynamics, with a focus on a beautiful interplay between algebraic geometry, combinatorics, dynamics, and number theory. 

  In this talk, we consider the so called sharp Hardy inequalities in a limiting case for   where  is an -dimensional ball.  We introduce a new transformation which relates the above inequality to the usual Hardy inequality in the whole space.
  As a result, we improve the sharp Hardy inequality in a limiting case on a ball by adding some remainder terms.

An Abelian differential defines a flat structure on the underlying Riemann surface, such that it can be realized as a plane polygon with edges suitably identified. Varying the shape of the polygon induces a GL(2,R)-action on the moduli space of Abelian differentials, which is called Teichmueller dynamics. In this lecture series, I will give an elementary introduction to Teichmueller dynamics, with a focus on a beautiful interplay between algebraic geometry, combinatorics, dynamics, and number theory. 

신호(음성 또는 영상)의 분석, 처리, 축약, 복원, 잡음제거, 특이신호의 감지 등에는 고전적으로 푸리에급수 또는 푸리에변환 등이 주로 많이 이용되어 왔다. 보다 국소적인 신호분석을 위한 노력으로 short-time Fourier 변환의 형태를 거처. 최근에는 웨이브릿 또는 가보 함수계를 이용한 신호(또는 함수)의 이산적인 표현 및 복원의 새로운 이론이 도입되어 널리 사용되고 있다. 1980년대 후반 I. Daubechies 에 의한 웨이브렛 직교계의 발견으로 웨이브렛의 이론과 응용의 연구가 폭발적으로 이루어 지고 있으며, Gabor 힘수계의 이론은 1946년의 D. Gabor 가 통신이론을 위하여 처음 제기되었으나, 웨이브렛 이론과 더불어 재조명되면서 활발히 연구가 수행되고 있다. 두 이론은 평행으로 전개되면서 최근 들어 근본적인 문제가 해결되고 재기되면서 응용조화해석(applied harmonic analysis)분야의 중심축을 이루고 있다. 이번 강연에서는 수학적인 면에서 기초적인 이론을 소개하여 수학과 학생들의 새로운 수학에 대한 이해를 돕고자 한다.

참석하고자 하시는 분은 아래 링크를 통해 사전등록 해주시면 감사하겠습니다^^
http://goo.gl/qMbQu2

일  시 : 3월 13일(금)  14:00 – 18:00

장  소 : 세미나실 E6-1 #1409

연  사 : 지 동 표 교수 (서울대학교)

강의내용 

1. Several cases where physics constrians computation,

      Several cases where computation constrains physics.

     Abstract: Computation can be thought as abstract process. But it is closely related to physics as our personal computer are fundamentally a physical device. Hence fundamental physics law might  constrain computation. Conversely complexity of computations gives constraints to  law  of physics.

We will discuss several cases of each without any technicality.

2. Information, entropy and all that

  Abstract: We will discuss about information theory from scratch and even will explain exotic  phenomena in quantum world

3. Quantum helps classical

    Abstract: We will several several mathematical facts which have nothing to do with quantum mechanics,

   but using quantum information theory  we can prove them much more  easily, or find better refined form of the statements.

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정확한 날씨예보 생산을 위한 예보관과 관측자료의 기여도가 각각 28%와 32%이나 수치예보모델이 미치는 영향은 40%라는 조사 결과가 있다. 기상청은 날씨예보 정확도 향상을 위하여 천리안 기상위성과 같은 관측 인프라 구축 뿐 아니라 수치예보모델 발전에 많은 투자를 하고 있다. 기상청 수치예보모델의 역사는 1989년 수치예보반을 설립하면서 처음 시작하였으며, 1995년 일본기상청의 전지구예보모델을 거쳐 영국의 통합모델을 도입 운영하고 있다. 기상청에서 운영하는 수치예보모델의 정확도는 세계 6위 또는 그 이상인 것으로 여러 비교 검증 결과가 보여주고 있으나 기상청은 이에 만족하지 않고 자체 선진 수치예보모델 기술 확보를 위하여 9년 동안 총 1000억원의 연구비를 투자하여 한국형수치예보모델 개발 사업을 추진 중에 있다. 수치예보모델의 문제 해결 및 발전을 위하여 응용수학과 같은 분야의 지식과 경험이 매우 필요하다. 이번 세미나에선 수치예보모델에 대한 응용수학 분야의 공동연구 과제 발굴에 필요한 기반 정보를 제공하기 위하여 기상청 수치예보모델의 현황과 문제점에 대해 발표할 예정이다.

An Abelian differential defines a flat structure on the underlying Riemann surface, such that it can be realized as a plane polygon with edges suitably identified. Varying the shape of the polygon induces a GL(2,R)-action on the moduli space of Abelian differentials, which is called Teichmueller dynamics. In this lecture series, I will give an elementary introduction to Teichmueller dynamics, with a focus on a beautiful interplay between algebraic geometry, combinatorics, dynamics, and number theory. 

An Abelian differential defines a flat structure on the underlying Riemann surface, such that it can be realized as a plane polygon with edges suitably identified. Varying the shape of the polygon induces a GL(2,R)-action on the moduli space of Abelian differentials, which is called Teichmueller dynamics. In this lecture series, I will give an elementary introduction to Teichmueller dynamics, with a focus on a beautiful interplay between algebraic geometry, combinatorics, dynamics, and number theory. 

대한민국 전자 산업은 수십년간 국가 경제 성장의 동력이 되며 성장해 왔다. 초기 일본업체로 부터 기술을 배우며, 때로는 copy도 하고, 가격경쟁력을 통해 성장을 하다, 세계최고 수준에 올라섰다. 그러나 지금은 apple, Google 등 SW기반의 Platform을 강점으로 하는 기업들과, 급속하게 성장하는 중국업체들 사이에서 어려운 경쟁을 해 나가고 있다. 세계 경제가 어려워지는 최근의 상황에서 돌파구는 무엇일까? 어떻게 해결해 나갈 수 있을까?

Cluster algebras were discovered by Fomin and Zelevinsky in 2001. Since then, they have been shown to be related to diverse areas of mathematics and physics such as Total positivity, Quiver representations, String theory, Statistical physics, Non-commutative geometry, Teichmüller theory, Hyperbolic geometry, Tropical geometry, KP solitons, Integrable systems, Quantum mechanics, Lie theory, Algebraic combinatorics, Number theory and Poisson geometry.
We explain these connections between various fields in elementary languages.