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Information Center for Mathematical Science

세미나

Information Center for Mathematical Science

세미나

2018 서울대학교 수리과학부 세미나

In this talk we consider inhomogeneous cubic-quintic NLS in space dimension d=3 d=3 :

2018 서울대학교 수리과학부 세미나
2018 서울대학교 수리과학부 세미나

The theory of Optimal Transport (OT) has developed rapidly during the last 30 years, partly because OT has constantly found applications in many areas, not only in mathematics, but also in economics and other social sciences. More recently a variant of OT, called Martingale Optimal Transport (MOT), has been introduced and extensively studied by mathematical finance community who observed the close connection of MOT with the model-independent robust option pricing and hedging. In this course we give a brief introduction of OT and MOT, from simple economic and financial motivations to some basic mathematical theory and open problems.

2018 서울대학교 수학강연회

2018 서울대학교 수학강연회

양자컴퓨터와 양자암호로 대표되는 양자기술의 응용이 대두되면서 그 기반이 되는 양자정보학에 대한 관심이 그 어느때보다 높다. 양자정보학은 양자상태의 정보량 및 양자상태를 전송할 때 전달되는 정보량을 분석하고자 하는 목표를 가지고 있는데 많은 경우 유한차원 힐버트 공간에서의 선형대수학으로 설명할 수 있다는 점 때문에 요구되는 수학적 난이도가 높지 않은 편이었다. 하지만 미해결 문제들이 쌓여가면서 순수수학의 여러분야에서 도구를 빌려오게 된 것이 최근의 조류중 하나이다. 이 강연에서는 함수해석학(functional analysis), 특히 바나흐 공간의 국소 이론(local theory of Banach spaces)과 작용소 대수(operator algebra)의 Connes embedding 예상이 결정적으로 작용하여 양자정보학의 중요한 문제가 해결 되었던 최근 사례를 살펴보고자 한다. 이 강연에서 양자정보학이나 함수해석학에 대한 사전지식은 필요하지 않다.

2018 서울대학교 수학강연회

Convergence of Fourier series and integrals is the most fundamental question in classical harmonic analysis from its beginning. In one dimension convergence in Lebesgue spaces is fairly well understood. However in higher dimensions the problem becomes more intriguing since there is no canonical way to sum (and integrate) Fourier series (and integrals, respectively), and convergence of the multidimensional Fourier series and integrals is related to complicated phenomena which can not be understood in perspective of convergence in one dimension. The Bochner-Riesz conjecture may be regarded as an attempt to understand multidimensional Fourier series and integrals. Even though the problem is settled in two dimensions, it remains open in higher dimensions. In this talk we review developments in the Bochner-Riesz conjecture and discuss its connection to the related problems such as the restriction and Kakeya conjectures.

2018 서울대학교 수학강연회

There have been several inequalities involving the volumes of convex bodies and of their dual objects or operations. We will go over basic concepts related to this topic and explain how the duality phenomena can be interpreted in various problems. Then we will talk about several volume inequalities arising from convex geometry and their possible generalizations.

2018 서울대학교 수학강연회

2018 서울대학교 수학강연회

The Lagrange spectrum is the set of approximation constants in the Diophantine approximation for badly approximated numbers. It is closely related with the Markov spectrum which corresponds the minimum values of indefinite quadratic forms over integral vectors. We discuss the Diophantine approximation for the rational points in the unit circle. We will introduce a dynamical system originally defined by Romik in 2008, study its Lagrange and Markov spectra.

2018 서울대학교 수학강연회

Lagrangian Floer theory in symplectic manifold associate a category (A infinity category) to a symplectic manifold. More than 20 years ago a relation of a relation between Lagrangian Floer theory and Gauge theory was studied by Floer himself various other people. A conjecture by Atiyah and Floer is based on this relation. One motivation of introducing A infinity category associated to a Symplectic manifold is to study Atiyah-Floer conjecture and related problems. In this talk, based on a joint work with A. Daemi, I want to explain some of the recent progress in this problem.

2018 서울대학교 수학강연회

In this talk, we will present a uniform dichotomy for generic GL(n,R) cocycles over a minimal base. We also apply it to study the spectra of discrete Schrodinger's operators. These are generalizations of results of Avila, Bochi and Damanik.

2018 서울대학교 수학강연회

We introduce the notion of congruences (modulo a prime number) between modular forms of different levels. One of the main questions is to show the existence of a certain newform of an expected level which is congruent to a given modular form. (For instance, if a given modular form comes from an elliptic curve over the field of rational numbers, then this problem is known as "epsilon-conjecture".) We partially answer this question when a given modular form is an Eisenstein series.

2018 서울대학교 수학강연회

A W-algebra is introduced as a symmetry algebra in 2-dimensional conformal field theory. Mathematical realization of a W-algebra was introduced by the theory of vertex algebras. Especially, W-algebras related to Lie superalgebras have been studied by many mathematicians and physicists. Moreover, classical W-algebras (classification of W-algebras) and finite W-algebras (finitization of W-algebras) are interesting objects in integrable systems theory and representation theories. In this talk, I will briefly introduce (classical, finite) W-algebras associated Lie superalgebras and related topics.

한국고등과학원 세미나

Gopakumar-Vafa correspondence relates the large N expansion of SU(N) HOMFLY invariants of the unknot in S^3, to topological strings on the resolved conifold. I will describe a generalization which relates HOMFLY invariants of analogs of the unknot in S^3/ADE, and topological strings on a non-toric CY3. The first ones have a matrix model description due to Marino, and the topological recursion on the matrix model spectral curve governs their large N expansion. We find by direct computation that the matrix model spectral curve is a 1-parameter specialization of the spectral curve of a relativistic Toda system of type ADE. This is consistent with the expectation from geometric engineering of a 5d gauge theory from the CY. The method we use applies in greater generality to compute the spectral curve of various matrix models. This is based on joint works with Eynard and Brini.

한국고등과학원 세미나

한국고등과학원 세미나

한국고등과학원 세미나

한국고등과학원 세미나

한국고등과학원 세미나

한국고등과학원 세미나

In this talk, we consider global solutions to the Ericksen-Leslie system modeling the hydrodynamic flow of nematic liquid crystals. In the presence of an applied magnetic field, the orientation of liquid crystals can be easily aligned along the direction of the external field. We discuss dynamical instabilities of global solutions to the Ericksen-Leslie system due to magnetic fields in dimension two. This is based on joint work with Yuan Chen and Yong Yu.