NamGyu Kang 
NamGyu Kang

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Professor in the School of Mathematics at Korea Institute for Advanced Study
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Analysis and Probability Seminar
09/21/2018 Ildoo Kim (Korea University): A regularity theory for degenerate equationsDiffusion equations is one of most famous partial differential equations (PDEs). Lots of generalized diffusion equations have appeared on the basis of scientific meaning. For instance, degenerate diffusion equations appeared long time ago due to observation of stopping diffusion intermittently. In this talk, we mostly introduce the reason why we study generalized diffusion equations far beyond the heat equation and we are going to discuss the change of regularity of solutions depending on the degeneracy of diffusion if time is allowed. 09/19/2018 Eunhee Jeong (KIAS): Bounds for the bilinear BochnerRiesz operatorThe bilinear BochnerRiesz operator is a bilinear multiplier operator related to convergence of the Fourier series. In this talk, we are concerned with boundedness of the bilinear BochnerRiesz operator on Lebesgue spaces. Especially, we focus on our recent result for the bilinear operator, which is a joint work with Sanghyuk Lee and Ana Vargas. We make use of a decomposition which relates the estimate for the bilinear BochnerRiesz operator to those of the square function for the classical BochnerRiesz operator. 09/13/2018 Sanghyuk Lee (SNU): Square function estimates for the BochnerRiesz meansIn this talk we are concerned with the squarefunction estimate associated with the BochnerRiesz means (also known as Stein's square function). Originally, the square function was introduced by Stein to study pointwise convergence of the BochnerRiesz means but it later turned out that the square function has important applications to other problems, such as $L^p$ boundedness for radial multipliers and smoothing properties for the Schrödinger and wave operators. We discuss the sharp $L^p$ estimate for the square function including the recent development based on multilinear (adjoint) restriction estimate due to Bennett, Carbery and Tao. 07/13/2018 Dongho Chae (ChungAng University): On the Type I singularity for the Euler equationsThe scenario of Type I singularity is a natural generalization of the selfsimilar or discretely selfsimilar singularities. Studying the problem of Type I singularities would be eventually helpful for future understanding of the possible singularities in the Euler system. In this talk, after short review of the studies of the selfsimilar solutions we survey recent works on scenario of the Type I blowup. By applying the local analysis methods, which have been useful in the elliptic or parabolic regularity theories, we could make some progresses in our study of the Type I blowup in the Euler equations. The project is jointly done with J. Wolf. 06/29/2018 Chanwoo Oh (UC Berkeley): Metastability of the Zero Range Process on a Finite Set Without Capacity Estimatesn this talk, I'll prove metastability of the zero range process on a finite set without using capacity estimates. The proof is based on the existence of certain auxiliary functions. One such function is inspired by Evans and Tabrizian's article, "Asymptotics for the KramersSmouchowski equations". This function is the solution of a certain equation involving the infinitesimal generator of the zero range process. Another relevant auxiliary function is from a work of Beltran and Landim. We also use martingale problems to characterize Markov processes. Let $p$ be the jump rates of a random walk on a finite set $S$. Assume that the uniform measure on $S$ is an invariant measure of this random walk for simplicity (we expect that our method is applicable for an arbitrary invariant measure $m$). Consider the zero range process on $S$, where the rate the particle jumps from a site $x$ to $y$ with $k$ particles at the site $x$ is given by $g(k)p(x,y)$. Here $g(0)=0, g(1)=1$, and $g(k)=(k/(k1))^\alpha$, $k>1$ for some $\alpha > 1$. As total number of particles $N \rightarrow \infty$, most of the particles concentrate on a single site. In the time scale $N^{1+\alpha}$, the site of concentration evolves as a Markov chain whose jump rates are proportional to the capacities of the underlying random walk. This talk based on the joint work with F. Rezakhanlou. 06/14/2018 Danny Nam (Princeton): Cutoff phenomenon for the SwendsenWang dynamicsThe SwendsenWang dynamics is an MCMC sampler of the Ising/Potts model, which recolors many vertices at once, as opposed to the classical singlesite Glauber dynamics. Although widely used in practice due to efficiency, the mixing time of the SwendsenWang dynamics is far from being wellunderstood, mainly because of its nonlocal behavior. In this talk, we prove cutoff phenomenon for the SwendsenWang dynamics on the lattice at high enough temperatures, meaning that the Markov chain exhibits a sharp transition from “unmixed” to “wellmixed.” The proof combines two earlier methods of proving cutoff, the update support [LubetzkySly ’13] and information percolation [LubetzkySly ’16], to establish cutoff in a nonlocal dynamics. Joint work with Allan Sly. 06/05/2018 SunSig Byun (SNU): Regularity theory for nonlinear elliptic and parabolic problemsWe discuss recent advances in the regularity estimates of solutions to nonlinear elliptic and parabolic problems. 05/25/2018 MoonJin Kang (Sookmyung Women's University): Uniqueness of entropy shocks to the isentropic Euler system in a class of inviscid limits from a large family of NavierStokes systemsWe briefly present issues on uniqueness of entropy solutions to the compressible Euler system, which are related to the long standing conjecture: The compressible Euler equations admit a unique entropy weak solution in a class of vanishing viscosity solutions as inviscid limits of solutions to the associated viscous system that is compressible NavierStokes system. As a study on the conjecture for an entropy shock, we prove the contraction property for any weak perturbations of viscous shocks of the barotropic NavierStokes systems with various viscosity coefficients, by constructing a weighted relative entropy and timedependent shift. The contraction property of the shocks does not depend on the viscosity coefficient. Therefore, this provides a weak compactness for the inviscid limit problem, that is, entropy shocks for the isentropic Euler system are stable and unique in the class of weak inviscid limits of solutions to the NavierStokes system. 05/16/2018 Kyeongsik Nam (UC Berkeley): Elliptic and hypoelliptic diffusions with singular driftAccording to the results by Krylov and Röckner, unique strong solution exists to stochastic differential equations with nondegenerate noise when the drift vector field belongs to $L^q([0,T],L^p_x)$ for $p,q\in (1,\infty)$ satisfying $\frac{2}{q}+\frac{d}{p}$ < 1. In this talk, we discuss two different directions to generalize the previous results: the critical case $\frac{2}{q}+\frac{d}{p}=1$, and the degenerate noise. I will introduce a Lorentz space to deal with the critical case and hypoelliptic theory to consider the degenerate case. Brief theory of the analysis on Lie groups will be covered and I will explain how this theory can be applied to prove the probabilistic results. 05/11/2018 Ji Oon Lee (KAIST): Behavior of the Spectral Edge in Random Matrix TheoryThe limiting eigenvalue statistics of a Wigner matrix is given by the semiclrcle law. For the semicircle law, the behavior of the spectral edge is characterized by the squareroot decay and the location of the edge. The behavior of the spectral edge for other random matrices can be understood by complex analytic method based on the Stiejtes transform. In this talk, we consider examples such as deformed Wigner matrices and sparse random matrices and analyze the behavior of the spectral edges of those random matrices. 05/04/2018 Joeun Jung (KIAS): Timefrequency analysis of Singular Fourier MultipliersWe will discuss about few types of multilinear operators with singular multipliers and their wide range of $L^p$ estimates by applying timefrequency analysis technique. 04/25/2018 InJee Jeong (KIAS): Singularity formation for the incompressible Euler equationsWe construct a local wellposedness class for the $3D$ incompressible axisymmetric Euler equations, and show that within this class there exist compactly supported initial data which blows up in finite time. This local wellposedness class is critical in terms of the natural scaling transformation of the Euler equations and consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of HouLuo, KiselevSverak, and many others, and the proof builds up on our previous works on $2D$ Euler and Boussinesq systems. An essential idea is the robustness of scaleinvariant dynamics for inviscid transport systems. This is joint work with Tarek Elgindi. 04/17/2018 Sanghyun Kim (SNU): Diffeomorphism groups of critical regularityFor each real number $a=k+s\ge1,$ where $k=[a],$ we define $Diff(R;a)$ as the group of compactly supported $C^k$diffeomorphisms of the real line whose $k$th derivatives are $s$Höldercontinuous. We prove that there exists a finitely generated group $G$ inside $Diff(R;a)$ such that $G$ admits no injective homomorphisms into the group $U \{ Diff(R;b) : b > a \}.$ The cases $a=0$ (Thurston; Calegari for $S^1$) and $a=1$ (Navas) are previously known. This is a joint work with Thomas Koberda. 12/13/2017 InJee Jeong (KIAS): A modification of the moment method and Plancherel random partitionsWe will begin by reviewing how a certain modification of the moment method was utilized to obtain limit theorems concerning fluctuations of extreme eigenvalues of various random matrix ensembles. Then we describe how to apply the method to analyze the asymptotic behavior of the first, second, and so on rows of stochastically decaying partitions. This is based on joint work with Sasha Sodin. 08/16/2017 Paul Jung (KAIST): Levy Khintchine random matrices and the Poisson weighted infinite skeleton tree
We study a class of Hermitian random matrices which includes Wigner matrices, heavytailed random matrices, and sparse random matrices such as adjacency matrices of ErdosRenyi graphs with $p=1/N.$
Our matrices have real entries which are i.i.d. up to symmetry.
The distribution of entries depends on $N,$ and we require sums of rows to converge in distribution; it is then wellknown that the limit must be infinitely divisible.
08/07/2017 Sanghyun Kim (SNU): Free products in Diff($S^1$)We prove that $(F_2 X Z) * Z$ does not embed into Diff$^{\,2}(S^1).$ Then we classify RAAGs that admit faithful smooth actions on the circle, answering a question raised in a paper of M. Kapovich. (Joint with T. Koberda) 03/30/2017 Sanghyun Kim (SNU): Flexibility of PSL(2,R) representationsWhich finitely presented groups arise as subgroups of PSL(2,R)? The general (indiscrete) case of this question is wideopen. We propose a class of torsionfree 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 KleinMaskit 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 PDEsSolutions 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 nonintermittent SPDEs (SPDEs with additive noise). Using the BarlowTaylor 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 ongoing work with Davar Khoshnevisan and Yimin Xiao. 08/03/2016 Jose Luis Romero (University of Vienna): Gabor analysis and sampling and interpolationGabor systems are structured function systems consisting of timefrequency 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 timefrequency 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 motionsConsider 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 coworkers 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 diffusionIn 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 curvatureWe 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 nonlocal operators in metric measure spacesIn this talk we consider a large class of nonlocal 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 nonlocal 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 TheoryThe 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 Sanghyun Kim (SNU): No finite index subgroups of mapping class groups act faithfully on the circle by $C^2$ diffeomorphismsOne 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 BurgerMonod theorem about obstructions of higher rank lattice actions on the circle. (Joint work with Hyungryul Baik and Thomas Koberda)
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Teaching at SNU
