We deal with special Hurwitz' schemes of curves, that is the coverings of the Riemann sphere having only odd ramification points. We will discuss the relation, considered firstly by Serre and Fried, with theta characteristic. In the line of a joint research project with G. Farkas and J. Naranjo we present some recent existence results in the particular case of elliptic and hyperelliptic curves.

 The Neumann-Poincare (NP) operator is a boundary integral operator which arises naturally when solving boundary value problems using layer potentials. It is not self-adjoint with the usual inner product. But it can symmetrized by introducing a new inner product on H^{-1/2} spaces using Plemelj's symmetrization principle. Recently many interesting spectral properties of the NP operator have been discovered. I will discuss about this development and various applications including solvability of PDEs with complex coefficients and plasmonic resonance.

Chemotaxis models are based on spatial or temporal gradient measurements
by individual organisms. The key contribution of Keller and Segel (J Theor
Biol 30:225–234, 1971a; J Theor Biol 30:235–248, 1971b) is showing that erratic
measurements of individuals may result in an accurate chemotaxis phenomenon as a
group. In this paper we provide another option to understand chemotactic behavior
when individuals do not sense the gradient of chemical concentration by any means.
We show that, if individuals increase their dispersal rate to find food when there is
not enough food, an accurate chemotactic behavior may be obtained without sensing
the gradient. Such a dispersal has been suggested by Cho and Kim (Bull Math Biol
75:845–870, 2013) and was called starvation driven diffusion. This model is surprisingly
similar to the original Keller–Segel model. A comprehensive picture of traveling
bands and fronts is provided.

 Let C be a smooth curve which is complete intersection of a quadric and a degree k>2 surface in the 3 dimensional projective space. Let C(2) be its second symmetric power. of C. We study the finite generation of the extended canonical ring R(Δ,K):=⨁(a,b)H^0(C(2),aΔ+bK), where Δ is the image of the diagonal and K is the canonical divisor. We show that R(Δ,K) is finitely generated if and only if the difference of the two linear series defined on C by the rulings of the quadric is a torsion non-trivial line bundle. Then we show that this holds on an analytically dense locus of the moduli space of such curves. The results have been obtained in a joint work with Antonio La Face and Michela Artebani.

The subject of this talk is wave equations that arise from geometric considerations. Prime examples include the wave map equation and the Yang-Mills equation on the Minkowski space. On the one hand, these are fundamental field theories arising in physics; on the other hand, they may be thought of as the hyperbolic analogues of the harmonic map and the elliptic Yang-Mills equations, which are interesting geometric PDEs on their own.

Our main concern will be global well-posedness for large data of these PDEs in dimensions where the conserved energy is critical with respect to the scaling symmetry of the equations. I will first explain the ‘threshold conjecture’ for wave maps and its resolution by Sterbenz-Tataru (cf. related work by Krieger-Schlag and Tao), as well as its latest refinement in my work with A. Lawrie. I will also describe my recent work with D. Tataru on the global well-posedness of the energy critical Maxwell-Klein-Gordon system, which shares many similarities with the Yang-Mills equation.

The subject of this talk is wave equations that arise from geometric considerations. Prime examples include the wave map equation and the Yang-Mills equation on the Minkowski space. On the one hand, these are fundamental field theories arising in physics; on the other hand, they may be thought of as the hyperbolic analogues of the harmonic map and the elliptic Yang-Mills equations, which are interesting geometric PDEs on their own.

Our main concern will be global well-posedness for large data of these PDEs in dimensions where the conserved energy is critical with respect to the scaling symmetry of the equations. I will first explain the ‘threshold conjecture’ for wave maps and its resolution by Sterbenz-Tataru (cf. related work by Krieger-Schlag and Tao), as well as its latest refinement in my work with A. Lawrie. I will also describe my recent work with D. Tataru on the global well-posedness of the energy critical Maxwell-Klein-Gordon system, which shares many similarities with the Yang-Mills equation.

We first investigate the Kronecker Ugendtraum (= Hilberet 12th Problem) which initiates the study of algebraic number theory. We also briefly review the class number one problem in terms of modular functions without using L-function arguments. And, over cyclotomic fields and imaginary biquadratic fields we show how to construct class fields(= abelian extensions) by making use of high dimensional modular functions.

We consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution can be arbitrary large. In this paper we derive an asymptotic formula of the solution, which characterizes the gradient blow-up of the solution in terms of conductivities of inclusions as well as the distance between inclusions. The asymptotic formula is expressed in bipolar coordinates in terms of the Lerch transcendent function, and it is valid for inclusions with arbitrary constant conductivities. We illustrate our results with numerical calculations.

If one were to write up a list of keywords that describe recent development in algebraic geometry, it would be hard to miss the words like "derived category" or "categorification" on the top part. One basic problem in algebraic geometry is to study how a variety can be embedded in other varieties. In 2011, Bondal categorified the embedding problem and raised the following question.

Question. (Fano visitor problem) Let Y be a smooth projective variety. Is there a Fano variety X equipped with a fully faithful embedding of the derived category of Y into that of X?
If there is such an X, then Y is called a Fano visitor and X a Fano host of Y. In this talk, I will talk about a joint work with In-Kyun Kim, Hwayoung Lee and Kyoung-Seog Lee in which we proved that every complete intersection is a Fano visitor. I will also discuss related questions and problems.

I will explain the basic structure theory and representation theory of reductive and semisimple algebraic groups, and illustrate an application to invariant theory. A connected reductive group is naturally a central extension of a connected semisimple group by a torus, which enables one to reduce problems about reductive groups to problems about semisimple groups and tori. I will apply this principle to the study of weight decompositions of representations and obtain a precise formula relating the states of reductive group actions and the states of their derived group actions.

To be rational, or not to be, that is the internal conflict a cubic faces; its linear destiny provides a way to rationalize its life in a rich world and its whimsical action shows a way for a cubic to refuse its rational life in an impoverished field.

Given a vector bundle E over a smooth scheme X, a classical result of Kempf-Laksov describes the Schubert classes of grassmann bundles Gr(d,E) by means of a Jacobi-Trudi determinant whose entries are polynomials in the Chern classes of E and the universal bundle U_d. More recently, using a similar geometric framework, Kazarian was able to obtain a Pfaffian formula describing the Schubert classes of the Lagrangian grassmann bundle. In this talk I will present how these determinantal and Pfaffian formulas can be generalized to connective K-theory, an oriented cohomology theory which can be specialized to both the Chow ring and the Grothendieck ring of vector bundles. This is a joint work with T. Ikeda, T. Matsumura, H. Naruse.

In his death bed letter to Hardy, Ramanujan introduced mock theta functions, which are now prototypes of mock modular forms. The coefficients of mock modular forms encode the number of certain combinatorial objects and we will discuss how mock modularity works to investigate arithmetic properties for these counting functions. On the other hand, generating functions for certain unimodal sequences are now becoming prototypes of quantum modular forms. We will discuss how they are related and what we expect from quantum modularity.

수학에서 형식적인 정의를 보면 전혀 관계가 없어 보이는 것이 어떤 경우 서로 밀접히 연관된 경우가 많다. 이런 관계성을 정확히 파악하고 증명하는 것이 수학의 중요한 일면일 것이다. PDE를 공부하다 보면 계산이 너무 많아 수학에서 추구하는 미적인 성향이 약하다고 생각하는 경향이 많다. 본 강연에서는 PDE분야에서 아름다운 결과 중 하나인 Symmetry에 관하여 별로 관련성이 없어 보이는 Maximum Principle을 통하여 보이는 방법에 대하여 소개하고자 한다.

 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):