The MathNet Korea
Information Center for Mathematical Science

세미나

Information Center for Mathematical Science

세미나

2018 서울대학교 수리과학부 세미나

The mini-course is an introductory and self-contained approach to the method of intrinsic scaling, aiming at bringing to light what is really essential in this powerful tool in the analysis of degenerate and singular equations. The theory is presented from scratch for the simplest model case of the degenerate p-Laplace equation, leaving aside technical renements needed to deal with more general situations. A striking feature of the method is its pervasiveness in terms of the applications and I hope to convince the audience of its strength as a systematic approach to regularity for an important and relevant class of nonlinear partial dierential equations. I will extensively follow my book 14 , with complements and extensions from a variety of sources (listed in references) 6,7,17

2018 서울대학교 수리과학부 세미나

In the second seminar we will continue to the case of non-compact Lie groups focusing on the specific examples of the Heisenberg group and the Euclidean motion group. Those groups are representative examples of nilpotent Lie groups and non-nilpotent solvable Lie groups, respectively. It turns out that the technical details for the two groups are quite different, so that it is too early to develop a general theory for all Lie groups at this moment. At the end of the seminar we will discuss some questions remained.

2018 서울대학교 수리과학부 세미나

The mini-course is an introductory and self-contained approach to the method of intrinsic scaling, aiming at bringing to light what is really essential in this powerful tool in the analysis of degenerate and singular equations. The theory is presented from scratch for the simplest model case of the degenerate p-Laplace equation, leaving aside technical renements needed to deal with more general situations. A striking feature of the method is its pervasiveness in terms of the applications and I hope to convince the audience of its strength as a systematic approach to regularity for an important and relevant class of nonlinear partial dierential equations. I will extensively follow my book 14 , with complements and extensions from a variety of sources (listed in references) 6,7,17

2018 서울대학교 수리과학부 세미나

The mini-course is an introductory and self-contained approach to the method of intrinsic scaling, aiming at bringing to light what is really essential in this powerful tool in the analysis of degenerate and singular equations. The theory is presented from scratch for the simplest model case of the degenerate p-Laplace equation, leaving aside technical renements needed to deal with more general situations. A striking feature of the method is its pervasiveness in terms of the applications and I hope to convince the audience of its strength as a systematic approach to regularity for an important and relevant class of nonlinear partial dierential equations. I will extensively follow my book 14 , with complements and extensions from a variety of sources (listed in references) 6,7,17

2018 서울대학교 수리과학부 세미나

Let G be the isometry group of (d+1)-dimensional hyperbolic space. A subgroup H of G is quasi-Fuchsian if H is a convex cocompact discrete subgroup of G and the limit set of H is homeomorphic to the (d-1)-dimensional sphere. In this talk, I will explain how to construct examples of quasi-Fuchsian groups of G which are not quasi-isometric to the hyperbolic d-space using the Tits-Vinberg representation of Coxeter groups. Joint work with Ludovic Marquis.

2018 서울대학교 수리과학부 세미나

Fourier algebras are preduals of group von Neumann algebras, whose Banach algebra structure contain all the information on the underlying locally compact group. For example, its Gelfand spectrum allows us to recover the topological structure of the underlying group. In this talk we will introduce a weighted version of Fourier algebras with the hope to obtain a different aspects of underlying groups through Gelfand spectra. It turns out that we can actually "detect" complexification structure when the group is a Lie group. We will cover the details of easily accessible cases, namely the case of compact Lie groups with the necessary preliminaries including some Lie theory terminologies in the first seminar. Among compact Lie groups we will examine the case of SU(n) in detail. At the end of the seminar we will address the technicality of defining the weighted Fourier algebra of general locally compact groups.

2018 서울대학교 수리과학부 세미나

Fourier algebras are preduals of group von Neumann algebras, whose Banach algebra structure contain all the information on the underlying locally compact group. For example, its Gelfand spectrum allows us to recover the topological structure of the underlying group. In this talk we will introduce a weighted version of Fourier algebras with the hope to obtain a different aspects of underlying groups through Gelfand spectra. It turns out that we can actually "detect" complexification structure when the group is a Lie group. We will cover the details of easily accessible cases, namely the case of compact Lie groups with the necessary preliminaries including some Lie theory terminologies in the first seminar. Among compact Lie groups we will examine the case of SU(n) in detail. At the end of the seminar we will address the technicality of defining the weighted Fourier algebra of general locally compact groups.

2018 서울대학교 수리과학부 세미나

We investigate the fundamental concepts such as delta hedging and Girsanov’s theorem in option pricing methods including binomial tree model, partial differential equation approach, and the martingale method, emphasizing rigorous theoretical foundation for binomial tree model based on the asymptotic martingale theory.

2018 서울대학교 수리과학부 세미나

Riley polynomial is defined for some kind of a presentation of 2-bridge link groups. Any zero of this polynomial is corresponding to a parabolic representation in SL(2;C). If there exists epimorphism between link groups, then Riley polynomial of the source can be divided by the one of the target.

2018 서울대학교 수리과학부 세미나

Reidemeister torsion is a numerical invariant for a finite cell complex with a linear representation of the fundamental group. For Brieskorn homology 3-spheres and the manifolds obtained by 1/n-surgery along the figure eight knot, we can give explicit formulas by using Chebyshev polynomials of the first and second types. Further we discuss the relation between this polynomial and the SL(2;C)-Casson invariant. This is part of joint works with Anh Tran.

2018 서울대학교 수리과학부 세미나

Twisted Alexander polynomial is one generalization of the Alexander polynomial for a link with linear representation. In this talk we consider this invariant for a SL(2;C)-representation and discuss the behavior of this for a torus link. This is part of joint works with Anh Tran and Takayuki Morifuji.

2018 서울대학교 수리과학부 세미나

The aim of this talk is to study Riesz transform associated to the Grushin operator. Since the Grushin operator is closely related to the Hermite opertor, we use the spectral decomposition of Hermite operator. The main theorem is L^p-boundedness of Riesz transform. To prove this, we use Littlewood-Palet theory and an operator-valued Fourier multiplier theorem due to L.Weis.

2018 서울대학교 수리과학부 세미나

The theory of interacting particle systems enables a rigorous analysis of several models that arise in statistical mechanics, biology, economy and other fields. We give an introduction to the theory of interacting particle systems. Our approach is based on the geometry and analysis on continuous configuration space in dimension d. As application we study a contact model in continuum, which describes the spread of an infectious disease. Technically, the contact process is a Markov process on a space of spin configurations. From the evolution of large, interacting particles systems (microscopic models) one can derive mean-field evolution equations. This aspect will also be discussed.

2018 서울대학교 수리과학부 세미나

In this talk, we will concentrate on special case of algebraic varieties, smooth projective complete intersections. We will construct a special type of homotopy algebra, called the DGBV (differential Gerstenhaber Batalin-Vilkovisky) algebra associated to any smooth projective complete intersections X. This DGBV algebra gives a new way to compute the primitive middle dimension cohomology and the period integrals of X. Moreover, this construction is purley algebraic and algorithmic which is computable using a computer program. As an application, we get an explicit algorithm to compute the inverse value of the modular j-fuction. The talk is based on joint work with Jeehoon Park.

2018 서울대학교 수리과학부 세미나

2018 서울대학교 수리과학부 세미나

In this talk we will give some boundedness results of pseudo-differential operators of type (0,0) on Triebel-Lizorkin spaces. We also discuss the sharpness of our results.

2018 서울대학교 수리과학부 세미나

Decomposing a function is widely used tool to deal with many situations. Among them, one important tool is decomposing some specific functions(Fourier supported in a neighborhood of some hyper-surfaces) with pieces of small supports in the Fourier side, rather than the physical side. After the decomposition, we need to control our original function with decomposed pieces. Precisely, we can estimate the L^p norm of the function with the square mean of decomposed pieces by using Cauchy-Schwarz inequality. However, Cauchy-Schwarz inequality makes large coefficient in front of the square mean. The conjecture is how small this coefficient can be. We will prove the l^2 decoupling conjecture for compact hyper-surfaces with positive definite second fundamental form. Although decoupling conjecture is a weaker version of the square function estimate problem, there are various applications such as discrete restriction phenomena, Strichartz estimates for torus, and some number theory problems. We will see how the decoupling conjecture used in these situations.

2018 서울대학교 수리과학부 세미나

In 1972 Galambos published an extreme value law for largest entries in continued fractions expansions. In fact, Doeblin had already proven a Poisson law for continued fractions in 1937, which implies the result of Galambos. But a gap was discovered in Doeblins proof and only filled around 1972 by Iosifescu. Interestingly, Iosifescu used aspects of Galambos' proof to fill this gap, hence all three mathematicians may reasonably be credited with the Poisson law.

2018 서울대학교 수리과학부 세미나

It is well known that AF-embeddable C*-algebras are quasidiagonal, and quasidiagonal C*-algebras are stably finite. Also for certain class of C*-algebras including graph C*-algebras and higher rank graph C*-algebras, these three properties are known to be equivalent while this is not the case in general. In this talk we discuss the properties with C*-algebras associated to labelled graphs.

2018 서울대학교 수리과학부 세미나

Decomposing a function is widely used tool to deal with many situations. Among them, one important tool is decomposing some specific functions(Fourier supported in a neighborhood of some hyper-surfaces) with pieces of small supports in the Fourier side, rather than the physical side. After the decomposition, we need to control our original function with decomposed pieces. Precisely, we can estimate the L^p norm of the function with the square mean of decomposed pieces by using Cauchy-Schwarz inequality. However, Cauchy-Schwarz inequality makes large coefficient in front of the square mean. The conjecture is how small this coefficient can be. We will prove the l^2 decoupling conjecture for compact hyper-surfaces with positive definite second fundamental form. Although decoupling conjecture is a weaker version of the square function estimate problem, there are various applications such as discrete restriction phenomena, Strichartz estimates for torus, and some number theory problems. We will see how the decoupling conjecture used in these situations.