강남규 교수 고등과학원 수학부
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Professor in the School of Mathematics at Korea Institute of Advanced Study
Seminars at KIAS
03/30/2017 Sang-hyun Kim (SNU): Flexibility of PSL(2,R) representations
Which finitely presented groups arise as subgroups of PSL(2,R)? The general (indiscrete) case of this question is wide-open. We propose a class of torsion-free groups, called "flexible groups"; these groups admit uncountably many, independent, indiscrete faithful representations into PSL(2,R). We prove a combination theorem for this class of groups. A key underlying idea is a version of Klein--Maskit combination theorem. This is a joint work with Thomas Koberda and Mahan MJ.
08/19/2016 Kunwoo Kim (Postech): Multifractal structure of the tall peaks for stochastic PDEs
Solutions to a large family of stochastic PDEs (SPDEs) exhibit tall peaks on some small regions and this phenomenon is called intermittency. In this talk, we consider the geometric structure of the tall peaks for two types of SPDEs: intermittent SPDEs (SPDEs with nonlinear noise) and non-intermittent SPDEs (SPDEs with additive noise). Using the Barlow-Taylor macroscopic Hausdorff dimension, we will show that there are infinitely many length scales in the tall peaks for both types of SPDEs. This is based on an on-going work with Davar Khoshnevisan and Yimin Xiao.
08/03/2016 Jose Luis Romero (University of Vienna): Gabor analysis and sampling and interpolation
Gabor systems are structured function systems consisting of time-frequency translates of a window function. I will discuss some of the central problems in Gabor analysis and progress towards them. In particular, I will present recent results on stability of the spanning properties of such systems under deformations of the underlying set of time-frequency nodes. The deformations that we consider are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. The proofs involve a characterization of Gabor frames and Gabor Riesz sequences in the style of Beurling's characterization of sets of sampling for bandlimited functions ('without inequalities').
07/25/2016 Insuk Seo (UC Berkeley): Limit theory and large deviation principle for interacting Brownian motions
Consider a system of interacting Brownian motions. Because of the complex nature of interactions among particles, it is difficult to obtain the typical or atypical behavior of a single particle. Varadhan and his co-workers have extensively studied such a problem for interacting particle systems on lattice. In the first part of the presentation, we review the theory proposed by Varadhan et al., which is now called the hydrodynamic limit theory. In the second part or the presentation, we explain our recent generalization of this theory to a system of interacting Brownian motions and discuss the universality of the results.
07/18/2016 Kijung Lee (Ajou University): Modeling: random interruptions to Heat diffusion
In the inhomogeneous heat equation $$u_t(t,x) = \Delta u(t,x)+f(t,x),$$ the term $f$ models the interruptions to the heat diffusion along time and on space locations. Especially, the effect of $f$ in time direction is more troublesome and the regularity of $u$ is subject to the regularity of $f.$ In this talk we discuss a type of modeling of $f$ in the form $N_t(\omega)g(t,x),$ where $N$ is a random noise process which can be whiter than the white noise. We also discuss a regularity relation between $N, g$ and $u.$
06/27/2016 Seonhee Lim (SNU): Asymptotics of heat kernel in Riemannian manifolds of negative curvature
We will show the asymptotics of the heat kernel $p(t,x,y)$ as $t$ goes to infinity for Riemannian manifolds of negative curvature. We will explain how to use dynamics of the geodesic flow and certain Gibbs measures. This is a joint work with F. Ledrappier.
03/22/2016 Kyeonghun Kim (Korea Univ.): An introduction to $L_p$-theory of stochastic PDEs
Stochastic partial differential equations (SPDEs) are differential equations which include the effects of random forces and environments.
03/16/2016 Panki Kim (SNU): Factorization of harmonic functions and Martin boundary for non-local operators in metric measure spaces
In this talk we consider a large class of non-local operators on metric measure spaces. We first show that a uniform and scale invariant boundary Harnack principle holds. Then we study Martin boundaries corresponding to non-local operators. If infinity is accessible from an open set D, then there is only one Martin boundary point of D associated with it, and this point is minimal. Under suitable further assumptions there is only one Martin boundary point associated with infinity, and that this point is minimal if and only if infinity is accessible from D. This is based on joint works with Renming Song and Zoran Vondracek.
03/14/2016 Ji Oon Lee (KAIST): Comparison Methods in Random Matrix Theory
The local eigenvalue statistics of a large class of random matrices exhibits the same limiting behaviour, and such property is known as the universality. A typical strategy to prove the universality is to first compute the local statistics for the simplest case, usually the Gaussian case, and compare the given model with the simple model. In this talk, I will explain the comparison methods and how we can apply them to prove universality for complicated models such as deformed Wigner matrices or sample covariance matrices with general population.
03/02/2016 Sang-hyun Kim (SNU): No finite index subgroups of mapping class groups act faithfully on the circle by $C^2$ diffeomorphisms
One intriguing direction of research in surface theory is the analogy between mapping class groups and higher rank lattices. However, current understanding of finite index subgroups of mapping class groups are still rudimentary. We prove the result in the title, which was originally asked by Farb, and which is analogous to Ghys and Burger--Monod theorem about obstructions of higher rank lattice actions on the circle. (Joint work with Hyungryul Baik and Thomas Koberda)
Teaching at SNU