Analysis, PDE & Probability Seminar

Analysis, PDE & Probability Seminar

2021. 12. 21. (Tue) 15:00-16:00 Room Online
Minjae Park (MIT)

Wilson loop expectations as sums over surfaces in 2D Yang-Mills theory

ABSTRACT. Although lattice Yang-Mills theory on ℤᵈ is easy to rigorously define, the construction of a satisfactory continuum theory on ℝᵈ is a major open problem when d ≥ 3. Such a theory should assign a Wilson loop expectation to each suitable collection ℒ of loops in ℝᵈ. One classical approach is to try to represent this expectation as a sum over surfaces with boundary ℒ. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities.
In this talk, we show how to make sense of Yang-Mills integrals as surface sums for d=2, where the continuum theory is already understood. We also obtain an alternative proof of the Makeenko-Migdal equation and a version of the Gross-Taylor expansion. Joint work with Joshua Pfeffer, Scott Sheffield, and Pu Yu.
Join Zoom Meeting https://us02web.zoom.us/j/5258205033?pwd=SW41OGFyaFBDeFgrcFhTTVBkckYwZz09
2021. 12. 17. (Fri) 15:00-16:00 Room Online
Minjae Park (MIT)

Variational formula for Liouville quantum gravity

ABSTRACT. Liouville quantum gravity (LQG) is a one-parameter family of random fractal surfaces conjectured to be the canonical continuous model of many important classes of discrete surfaces in statistical physics, string theory, and conformal field theory. In probability theory, the γ-LQG measure is rigorously defined in terms of regularizations, but the original motivation of LQG surfaces in string theory is based on the idea of weighting a manifold by some power of the “determinant of Laplacian”.
We construct a one-parameter family of approximate LQG surfaces justifying the heuristic definition in physics during the talk. For this, we derive a variant of the Polyakov-Alvarez variational formula using Brownian loops in non-smooth setting. Joint work with Morris Ang, Joshua Pfeffer, and Scott Sheffield.
Join Zoom Meeting https://us02web.zoom.us/j/5258205033?pwd=SW41OGFyaFBDeFgrcFhTTVBkckYwZz09
2021. 12. 16. (Thu) 15:00-17:00 Room Online
Seong-Mi Seo (KAIST)

Analysis of the sequence of scalar Riemann-Hilbert problems

ABSTRACT. In the series of seminars, I will explain the method to obtain the universality of edge scaling limits of 2D determinantal Coulomb gas using asymptotic expansions of planar orthogonal polynomials. The method is based on the derivation of the Riemann-Hilbert hierarchy and the construction of an orthogonal foliation flow, which is introduced by Hedenmalm and Wennman.
In this talk, I will discuss the analysis of the sequence of scalar Riemann-Hilbert problems which gives an algorithm to obtain all higher order correction terms.
2021. 12. 8. (Wed) 14:00-16:00 Room Online
Kyeongsik Nam (KAIST)

Universal scaling limit of random planar maps

ABSTRACT. Liouville quantum gravity (LQG) is a random fractal surface constructed from Gaussian free field (GFF), first appeared in the physics literature. LQG surface has a great importance since it describes the universal large-scale behavior of discrete random geometries. Unlike smooth two-dimensional Riemannian manifolds, LQG surfaces are too rough and possess fractal properties. In this series of talks, I will give an overview of the LQG surface and talk about its probabilistic and geometric properties.
LQG metric space describes the universal scaling limit of random planar maps. I will briefly mention this connection and introduce some open problems. Based on the ongoing work with Shirshendu Ganguly.
Zoom link : https://us02web.zoom.us/j/86710448009?pwd=ckpVN2xXalY2b1libjJETFFVT2FrQT09
2021. 12. 3. (Fri) 15:00-17:00 Room Online
Kyeongsik Nam (KAIST)

Random metric properties of Liouville quantum gravity

ABSTRACT. Liouville quantum gravity (LQG) is a random fractal surface constructed from Gaussian free field (GFF), first appeared in the physics literature. LQG surface has a great importance since it describes the universal large-scale behavior of discrete random geometries. Unlike smooth two-dimensional Riemannian manifolds, LQG surfaces are too rough and possess fractal properties. In this series of talks, I will give an overview of the LQG surface and talk about its probabilistic and geometric properties.
Although the measure on the LQG surface has been well-understood, it is extremely delicate to study its metric properties. Very recently, a canonical metric associated with the LQG surface was successfully constructed. I will briefly talk about the construction of this random metric and its geometric properties, such as behaviors of geodesics.
Zoom link : https://us02web.zoom.us/j/86710448009?pwd=ckpVN2xXalY2b1libjJETFFVT2FrQT09
2021. 12. 2. (Thu) 16:00-18:00 Room Online
Seong-Mi Seo (KAIST)

Soft Riemann-Hilbert problem for orthogonal polynomials

ABSTRACT. In the series of seminars, I will explain the method to obtain the universality of edge scaling limits of 2D determinantal Coulomb gas using asymptotic expansions of planar orthogonal polynomials. The method is based on the derivation of the Riemann-Hilbert hierarchy and the construction of an orthogonal foliation flow, which is introduced by Hedenmalm and Wennman.
In this talk, I will introduce a soft Riemann-Hilbert problem for orthogonal polynomials, recently studied by Hedenmalm, which gives a simple algebraic framework for the terms of the asymptotic expansion.
2021. 11. 30. (Tue) 17:00-18:00 Room Online
Christophe Charlier (University of Copenhagen)

[GS_M_APP] Planar jump-type singularities and gap probabilities in the random normal matrix model

ABSTRACT. In this talk I will consider a two-dimensional point process which generalizes the Ginibre point process and which arises in the study of certain random normal matrices. I will discuss two problems: (1) obtain asymptotics for the generating function of the disk counting statistics, and (2) determine the large gap asymptotics when the hole region is rotation-invariant. Both these problems can be formulated as asymptotic problems about determinants with planar discontinuities (but the nature of these discontinuities are drastically different in problems (1) and (2)).
2021. 11. 30. (Tue) 15:00-17:00 Room Online
Kyeongsik Nam (KAIST)

Gaussian multiplicative chaos

ABSTRACT. Liouville quantum gravity (LQG) is a random fractal surface constructed from Gaussian free field (GFF), first appeared in the physics literature. LQG surface has a great importance since it describes the universal large-scale behavior of discrete random geometries. Unlike smooth two-dimensional Riemannian manifolds, LQG surfaces are too rough and possess fractal properties. In this series of talks, I will give an overview of the LQG surface and talk about its probabilistic and geometric properties.
LQG is a random surface whose metric tensor is expressed as an exponential of GFF. It naturally arises in the scaling limit of thick points of GFF. I will briefly talk about this connection and review Kahane's theory of Gaussian multiplicative chaos.
Zoom link : https://us02web.zoom.us/j/86710448009?pwd=ckpVN2xXalY2b1libjJETFFVT2FrQT09
2021. 11. 25. (Thu) 16:00-18:00 Room Online
Seong-Mi Seo (KAIST)

Hard edge scaling limits of 2D Coulomb gas system

ABSTRACT. In the series of seminars, I will explain the method to obtain the universality of edge scaling limits of 2D determinantal Coulomb gas using asymptotic expansions of planar orthogonal polynomials. The method is based on the derivation of the Riemann-Hilbert hierarchy and the construction of an orthogonal foliation flow, which is introduced by Hedenmalm and Wennman.
In this talk, I will focus on the orthogonal foliation flow for hard edge orthogonal polynomials and explain the hard edge scaling limits of 2D Coulomb gas system.
2021. 11. 24. (Wed) 15:00-1700 Room Online
Kyeongsik Nam (KAIST)

Liouville quantum gravity surfaces

ABSTRACT. Liouville quantum gravity (LQG) is a random fractal surface constructed from Gaussian free field (GFF), first appeared in the physics literature. LQG surface has a great importance since it describes the universal large-scale behavior of discrete random geometries. Unlike smooth two-dimensional Riemannian manifolds, LQG surfaces are too rough and possess fractal properties. In this series of talks, I will give an overview of the LQG surface and talk about its probabilistic and geometric properties.
GFF is a log-correlated Gaussian field, which is a central model of random height functions. I will briefly mention some key properties of GFF and talk about extreme value theory.
Zoom link : https://us02web.zoom.us/j/86710448009?pwd=ckpVN2xXalY2b1libjJETFFVT2FrQT09
2021. 11. 18. (Thu) 16:00-18:00 Room Online
Seong-Mi Seo (KAIST)

Planar orthogonal polynomials and edge universality for 2D Coulomb gases 2

ABSTRACT. In the series of seminars, I will explain the method to obtain the universality of edge scaling limits of 2D determinantal Coulomb gas using asymptotic expansions of planar orthogonal polynomials. The method is based on the derivation of the Riemann-Hilbert hierarchy and the construction of an orthogonal foliation flow, which is introduced by Hedenmalm and Wennman.
In this talk, I will explain on the construction of the orthogonal foliation flow, which is a flow of closed curves on which the quasi-orthogonal polynomials are asymptotically orthogonal.
2021. 11. 17. (Wed) 16:00-17:00 Room Online
Jaewhi Park (KAIST)

[GS_M_APP] Tracy-Widom limit for free sum of random matrices

ABSTRACT. In random matrix theory, local statistics of the eigenvalues of random matrices have been studied extensively. Among the flourishing researches, one of the most remarkable concepts in random matrix theory is the universality phenomenon. The universality implies that the local eigenvalue statistics depend only on the symmetric class of the matrix ensembles.
In this talk, I will explain the Tracy-Widom limit for the largest eigenvalue of the sum of two unitarily invariant matrices establishing the edge universality of the free sum of random matrices.
The proof is based on the Green function comparison method and local statistics of Dyson Brownian motion, a continuous flow of random matrix, starting from the original matrix. I will also explain the free additive convolution of probability measures, analytic subordination system of the convolution and its stability.
This talk is based on a joint work with Hong Chang Ji.
2021. 11. 11. (Thu) 16:00-18:00 Room Online
Seong-Mi Seo (KAIST)

Planar orthogonal polynomials and edge universality for 2D Coulomb gases 1

ABSTRACT. In the series of seminars, I will explain the method to obtain the universality of edge scaling limits of 2D determinantal Coulomb gas using asymptotic expansions of planar orthogonal polynomials. The method is based on the derivation of the Riemann-Hilbert hierarchy and the construction of an orthogonal foliation flow, which is introduced by Hedenmalm and Wennman.
In this talk, I will introduce the asymptotic expansion of planar orthogonal polynomials in exponentially varying weights and quasi orthogonal polynomial approximations.
2021. 11. 10. (Wed) 17:00-18:00 Room Online
Nicholas Simm (University of Sussex)

[GS_M_APP] Fluctuations and correlations for products of real asymmetric random matrices

ABSTRACT. Consider a finite product of real asymmetric N x N random matrices with i.i.d. Gaussian entries. Recently, Forrester and Ipsen obtained finite-N formulae for the correlation functions of the real eigenvalues of such products in terms of a Pfaffian point process. I will discuss recent work where we obtain asymptotic estimates for the correlation kernel of the process as N tends to infinity, in particular establishing universality of the real eigenvalues in the bulk and spectral edge regimes. I will explain how we apply such estimates to prove central limit theorems for linear statistics and to establish universality of the largest real eigenvalue. Joint work with Will FitzGerald (University of Manchester).
2021. 10. 29 (Fri) 11:00-12:00 Room Online
Thomas Alberts (University of Utah)

A Fixed Point Formula of a Random Dynamical System

ABSTRACT. I will discuss a particular dynamical system on the space of probability measures on the line and its relation to electrical networks. This system has powerful contractive properties that ensure it converges almost surely to a delta mass, with very few assumptions on the choice of the initial probability measure. However this is guaranteed by non-constructive arguments, and very little is known about how the location of the limiting delta mass depends on the initial condition. I will describe some recent solutions to this problem, for certain types of initial conditions, based on the connection with electrical network theory.
2021. 10. 27. (Wed) 14:30-15:30 Room Online
Ji Oon Lee (KAIST)

Local law and Tracy-Widom limit for sparse random matrices 4

ABSTRACT. For many random matrix models, one of the fundamental inputs in the proof of universality results is the local law, which is an estimate of the local eigenvalue density. In this series of seminars, I will introduce the method based on recursive moment estimates and cumulant expansions. I will also explain how the local law and the Tracy-Widom limit for the largest eigenvalue can be proved, where the main example is a sparse random matrix.
In this talk, I will introduce the proof of the Tracy-Widom limit for sparse random matrices. The focus will be on the Green function comparison method.
2021. 10. 27. (Wed) 11:00-12:00 Room Online
Thomas Alberts (University of Utah)

Uniform Spanning Trees and the Burton-Pemantle Theorem

ABSTRACT. This talk will review the uniform spanning tree model on general graphs, concentrating on its determinantal nature. In particular, I will review the celebrated Burton-Pemantle theorem, a fundamental and unexpected result which showed that its edge set is a determinantal point process, almost 150 years after Kirchoff first showed the basic version of this result for individual edges. Recent work on random matrix theory connected to electrical networks should shed some light on another method of analyzing this celebrated result, which I will discuss.
2021. 10. 25. (Mon) 11:00-12:00 Room Online
Thomas Alberts (University of Utah)

Random Matrix Theory for Composites on Graphs

ABSTRACT. I will discuss the random matrix theory behind two-component random resistor networks on general graphs. This involves random submatrices of the graph's natural Gamma projection operator, which is related to the gradient of the Gaussian Free Field on the graph. The particular realization of the submatrix is determined by the disorder of the conductances. Certain combinations of graph symmetries together with different models for the random conductances lead to exactly computable spectral statistics. Our recent results lead to exact spectral statistics for the uniform spanning tree model on a diamond hierarchical lattice. Joint work with Ken Golden, Elena Cherkaev, Ben Murphy, Han Le.
2021. 10. 19. (Tue) 14:00-16:00 Room 1424
Jaehyeon Ryu (Jeonbuk National University)

Dispersive estimates for principally normal pseudodifferential operators II

ABSTRACT. In this seminar, we review some results in the work of Koch and Tataru [1]. We study dispersive estimates for operators with real symbols contained in $\lambda S_{\lambda}^2$. Overall strategy for the proof and the key ingredients such as parametrix construction will also be discussed in detail.
Day 1(Monday) : We begin by introducing the symbol class $\lambda S_{\lambda}^2$ and some geometrical assumptions that the symbols should satisfy. Then we present the main theorem ([1, Theorem 2.5]), and thereafter study a reduction step of the proof.
Day 2(Tuesday) : We shall look into the core part of the proof. First we obtain explicit kernel representation of the fundamental solution operator $\partial_t + a^w(x,D)$ using the FBI transform. This serves a key tool in establishing a pointwise kernel estimate with a fixed time. Combining the estimate with a trivial $L^2$ bound, we can derive the mixed norm estimate, which yields desired results in the main theorem.
References
[1] H. Koch, and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math. 58 (2005), 217-284.
2021. 10. 18. (Mon) 14:00-16:00 Room 1423
Jaehyeon Ryu (Jeonbuk National University)

Dispersive estimates for principally normal pseudodifferential operators I

ABSTRACT. In this seminar, we review some results in the work of Koch and Tataru [1]. We study dispersive estimates for operators with real symbols contained in $\lambda S_{\lambda}^2$. Overall strategy for the proof and the key ingredients such as parametrix construction will also be discussed in detail.
Day 1(Monday) : We begin by introducing the symbol class $\lambda S_{\lambda}^2$ and some geometrical assumptions that the symbols should satisfy. Then we present the main theorem ([1, Theorem 2.5]), and thereafter study a reduction step of the proof.
Day 2(Tuesday) : We shall look into the core part of the proof. First we obtain explicit kernel representation of the fundamental solution operator $\partial_t + a^w(x,D)$ using the FBI transform. This serves a key tool in establishing a pointwise kernel estimate with a fixed time. Combining the estimate with a trivial $L^2$ bound, we can derive the mixed norm estimate, which yields desired results in the main theorem.
References
[1] H. Koch, and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math. 58 (2005), 217-284.
2021. 10. 13. (Wed) 14:30-15:30 Room Online
Ji Oon Lee (KAIST)

Local law and Tracy-Widom limit for sparse random matrices 3

ABSTRACT. For many random matrix models, one of the fundamental inputs in the proof of universality results is the local law, which is an estimate of the local eigenvalue density. In this series of seminars, I will introduce the method based on recursive moment estimates and cumulant expansions. I will also explain how the local law and the Tracy-Widom limit for the largest eigenvalue can be proved, where the main example is a sparse random matrix.
In this talk, I will extend the proof of the local law from the GOE case to general cases. I will focus on the cumulant expansion, which is also known as the generalized Stein lemma.
2021. 10. 6. (Wed) 14:30-15:30 Room Online
Ji Oon Lee (KAIST)

Local law and Tracy-Widom limit for sparse random matrices 2

ABSTRACT. For many random matrix models, one of the fundamental inputs in the proof of universality results is the local law, which is an estimate of the local eigenvalue density. In this series of seminars, I will introduce the method based on recursive moment estimates and cumulant expansions. I will also explain how the local law and the Tracy-Widom limit for the largest eigenvalue can be proved, where the main example is a sparse random matrix.
In this talk, I will introduce a local law for Gaussian Orthogonal Ensemble and its proof based on the recursive moment estimate. The difference between the averaged local law and the entrywise local law will also be explained.
2021. 9. 29. (Wed) 14:30-15:30 Room Online
Ji Oon Lee (KAIST)

Local law and Tracy-Widom limit for sparse random matrices 1

ABSTRACT. For many random matrix models, one of the fundamental inputs in the proof of universality results is the local law, which is an estimate of the local eigenvalue density. In this series of seminars, I will introduce the method based on recursive moment estimates and cumulant expansions. I will also explain how the local law and the Tracy-Widom limit for the largest eigenvalue can be proved, where the main example is a sparse random matrix.
In this talk, I will introduce the proof of the Tracy-Widom limit for Wigner matrices and explain the difference between Wigner matrices and sparse random matrices. The focus will be on the local law and several other important ingredients of the proof of the Tracy-Widom limit, such as the square-root decay at the edge and an upper bound on the largest eigenvalue.
2021. 9. 16. (Thu) 14:00-15:30 Room Online
Jinyeop Lee (Ludwig Maximilian University of Munich)

Convergence towards the Vlasov-Poisson Equation from the N-Fermionic Schrödinger Equation

ABSTRACT. We consider the quantum dynamics of N interacting fermions in the large N limit. The particles in the system interact with each other via repulsive interaction that is regularized Coulomb potential with a polynomial cutoff with respect to N. From the quantum system, we derive the Vlasov-Poisson system by simultaneously estimating the semiclassical and mean-field residues in terms of the Husimi measure.
2021. 9. 9. (Thu) 10:00-11:00 Room Online
Gregory Lawler (Chicago University)

An introduction to the Conformally Invariant Two-Dimensional World

ABSTRACT. This will be an expository series of talks introducing some related ideas arising in statistical physics in two dimensions: Gaussian free field (GFF), Schramm - Loewner evolution(SLE), loop measures and determinant of the Laplacian, Liouville quantum gravity(LQG), Minkowski content, and other fields and fractal measures. We will consider both discrete and continuous models. The talks will be intended primarily for those who do not know what these words mean.
Zoom link: https://us02web.zoom.us/j/85281739365
2021. 9. 8. (Wed) 10:00-11:00 Room Online
Gregory Lawler (Chicago University)

An introduction to the Conformally Invariant Two-Dimensional World

ABSTRACT. This will be an expository series of talks introducing some related ideas arising in statistical physics in two dimensions: Gaussian free field (GFF), Schramm - Loewner evolution(SLE), loop measures and determinant of the Laplacian, Liouville quantum gravity(LQG), Minkowski content, and other fields and fractal measures. We will consider both discrete and continuous models. The talks will be intended primarily for those who do not know what these words mean.
Zoom link: https://us02web.zoom.us/j/85281739365
2021. 9. 3. (Fri) 11:00-12:30 Room Online
Jihoon Ok (Sogang University)

Local Hölder regularity for nonlocal equations with nonstandard growth and differentiability

ABSTRACT. In this talk, we discuss local regularity for weak solutions to nonlocal problems. I introduce nonlocal equations of the fractional $p$-Laplacian type in the context of the calculus of variations and known regularity results for these equations. Finally, I also present our recent results about the local boundedness and the local Hölder continuity of weak solutions to nonlocal equations with nonstandard growth and differentiability under sharp assumptions.
Zoom link: https://us02web.zoom.us/j/85151918173?pwd=MCtaTTAzUktZOXVpaFlYdHpBcU5SUT09
2021. 9. 2. (Thu) 16:00-17:30 Room Online
In-Jee Jeong (Seoul National University)

Dynamics of vortex rings

ABSTRACT. We consider axisymmetric fluid flows, which could be considered as the simplest three-dimensional flows. Axisymmetric vortex rings with thin sectional radius could be considered as 3D analogues of point vortices. We present some rigorous results related to the dynamics of such vortex configurations. Finally we consider the case of antiparallel vortex rings, which is particularly interesting due to the possibility of infinite vortex stretching as the rings stretch out in the radial direction.
2021. 9. 1. (Wed) 16:00-17:30 Room Online
In-Jee Jeong (Seoul National University)

Existence, stability, and instability of 2D vortex equilibria

ABSTRACT. Persistence of large scale vortex structures in two-dimensional flows strongly suggest their stability. We look at some sources of dynamical stability: conservation laws, variational principles, and incompressibility. Furthermore, variational methods can be applied to produce non-trivial steady or traveling-wave solutions. Lastly, we discuss how stability can be used to prove instability, or small scale creation, for perturbations.
2021. 8. 31. (Tue) 16:00-17:30 Room Online
In-Jee Jeong (Seoul National University)

Point vortex motion in two-dimensional fluids

ABSTRACT. A drastic simplification of the two-dimensional incompressible fluid motion is provided by the point vortex dynamics. In this viewpoint, the vorticity of the fluid is given by a finite number of dirac deltas with different strengths. Then the dynamics is given explicitly by a system of ODEs governing the location of the point vortices. Some interesting long-time dynamics can be seen even in the completely integrable cases. We then present some classical results connecting point vortex dynamics with actual solutions of the incompressible fluid equation.
2021. 8. 30. (Mon) 16:00-17:30 Room Online
In-Jee Jeong (Seoul National University)

Incompressible fluids: open problems and classical theory

ABSTRACT. We introduce the incompressible Euler and Navier-Stokes equations, which describe the evolution of volume preserving fluids. The issue of long-time dynamics remains largely open even in the much simpler case of two-dimensional fluids. We present a few relevant concrete open problems, and discuss classical well-posedness theory: global existence of strong and weak solutions.
2021. 8. 27. (Fri) 17:00-18:30 Room Online
Wontae Kim (Seoul National University)

Self-improving property of very weak solution for the elliptic double-phase system

ABSTRACT. In this talk, we consider the very weak solution for the elliptic double-phase system. In particular, we shall discuss the technique of Lipschitz truncation for the elliptic double-phase system. Along with the finer estimate on Lipschitz truncated function, some very weak solution becomes a weak solution for the referenced system. This is joint work with Sumiya Baasandorj and Sun-Sig Byun.
Zoom link: https://us02web.zoom.us/j/81736334853?pwd=c2pQYjhoTzRoODBweHdZenI0N3YwUT09
2021. 8. 27. (Fri) 11:00-12:30 Room Online
Seungjin Ryu (University of Seoul)

Nonlinear parabolic double obstacle problems

ABSTRACT. In this talk, we are concerned with variational inequalities for p-Laplacian type parabolic equations with irregular double obstacles. The values of solutions under consideration are required to have two-sided constraints. We discuss parabolic double obstacle problems to establish the global Calderon-Zygmund type estimates for such solutions. In particular, the existence of such a solution shall be dealt with carefully as this solution is prescribed to satisfy the associated parabolic variational inequality instead of the parabolic equation.
Zoom link: https://us02web.zoom.us/j/85151918173?pwd=MCtaTTAzUktZOXVpaFlYdHpBcU5SUT09
2021. 8. 20. (Fri) 17:00-18:30 Room Online
Mikyoung Lee (Pusan National University)

$L^q$-regularity for nonlinear elliptic equations with Schrödinger-type inhomogeneous lower order terms

ABSTRACT. In this talk, we consider elliptic equations of the $p$-Laplacian type with inhomogeneous lower order terms which involve nonnegative potentials satisfying a reverse Hölder type condition. We discuss interior and boundary $L^q$ estimates for the gradient of weak solutions and the lower order terms, independently, under sharp regularity conditions on the coefficients and the boundaries.
Zoom link : https://us02web.zoom.us/j/81736334853?pwd=c2pQYjhoTzRoODBweHdZenI0N3YwUT09
2021. 8. 20. (Fri) 11:00-12:30 Room Online
Jeongmin Han (Seoul National University)

On some results of time-dependent tug-of-war games

ABSTRACT. In this talk, we are concerned with a stochastic game called ‘tug-of-war’. This game is considered as a discrete scheme of $p$-Laplace type operators. Since the study began in the late 2000s, there have been many advances in this field. We will introduce some results of value functions for time-dependent tug-of-war, such as regularity, asymptotic behaviors and relation with $p$-Laplace type equations.
Zoom link : https://us02web.zoom.us/j/85151918173?pwd=MCtaTTAzUktZOXVpaFlYdHpBcU5SUT09
2021. 8. 19. (Thu) 18:00-19:00 Room Online
Woojoo Shim (Seoul National University)

Euler-type discretization of thermodynamic Cucker-smale model on complete Riemannian manifolds

ABSTRACT. In this talk, we discuss an Euler-type discretization of second order ODE models on connected, complete and smooth Riemannian manifolds, and study its application to thermodynamic Cucker-smale(TCS) model. Our proposed discrete model is expressed in terms of an exponential map on a tangent bundle endowed with the Sasaki metric. Compared to projection-based discretization on manifolds, it is embedding free and enjoys the same structural properties as the corresponding continuous models. For the proposed model, we provide a sufficient framework leading to asymptotic velocity alignment of TCS particles. For the unit-d sphere (Sd), we provide explicit representations of the Sasaki metric and the corresponding geodesics on TSd, and show that the TCS model exhibits a dichotomy in asymptotic spatial patterns.
Zoom link : https://us02web.zoom.us/j/88624015514?pwd=UnA1UW5RWCtQTEEwNWJVOGRobnZ3UT09
2021. 8. 17. (Tue) 17:00-18:00 Room Online
Jonas Jalowy (University of Muenster)

[GS_M_APP] Fluctuations of the magnetization in the Block Potts Model

ABSTRACT. This talk will be about the block spin mean-field Potts model, in which the spins are divided into s blocks and can take q>1 different values (colors). Each block is allowed to contain a different proportion of vertices and behaves itself like a mean-field Ising/Potts model which also interacts with other blocks according to different temperatures. Of particular interest is the behavior of the magnetization, which counts the number of colors appearing in the distinct blocks. I will present central limit theorems for the magnetization in the generalized high temperature regime and show a moderate deviation principle for its fluctuations on lower scalings. We shall also compare our results to various related models and look at simulations that illustrate the results and interesting unsolved phenomena. This is joint work with Matthias Löwe and Holger Sambale.
2021. 8. 17. (Tue) 14:00-15:00 Room Online
Dongnam Ko (Catholic University of Korea)

Practical synchronization of Winfree oscillators in a random environment

ABSTRACT. We estimate emergent dynamics of Winfree oscillators under noise as in a heat bath. In a large coupling regime, The oscillators start accumulating in a small area in a short time while they spread by heat in long-time horizon, which is called a practical synchronization. Without noise, the deterministic Winfree model exhibits the oscillator death, emerging with a convergence of the phase ensemble. The additive noise, however, is expected to destroy the stability of the equilibrium. In this talk, by tracking the running maximum of the phase processes, we estimate the escaping probability from a small space interval near the equilibrium. This result quantitatively explains the robustness of the practical synchronization, which indicates that the finite-time emergent behavior from finite oscillators is close to the deterministic dynamics, known as the large deviation theory. Our approach produces explicit bounds on probabilities, relying on comparisons with the Ornstein-Uhlenbeck processes. It is hence optimal in the sense that the linearized model gives the same order.
Zoom link : https://us02web.zoom.us/meeting/register/tZAscu-trjwuHdztqmoLpV77UOegIGAwTue2
2021. 8. 6. (Fri) 17:00-18:30 Room Online
Yeonghun Youn (Yeungnam University)

Potential estimates for nonlinear elliptic measure data problems with obstacle

ABSTRACT. In this talk, we study the recent progress in the regularity theory for nonlinear elliptic measure data problems with obstacles and some difficulties arising from obtaining the gradient Riesz potential estimates for such problems. The interaction between the right-hand side and the obstacles makes it hard to get optimal linearized comparison estimates and potential estimates. We will see some methods to overcome such difficulties in this talk.
Zoom link : https://us02web.zoom.us/j/81736334853?pwd=c2pQYjhoTzRoODBweHdZenI0N3YwUT09
2021. 8. 6. (Fri) 11:00-12:30 Room Online
Youchan Kim (University of Seoul)

Piecewise smoothness for linear elliptic systems with piecewise smooth coefficients

ABSTRACT. Li and Vogelius in 2000 and Li and Nirenberg in 2003 showed piecewise $C^{1,\gamma}$ regularity for linear elliptic problems with piecewise $C^{\gamma}$ coefficients which come from composite materials.
In this talk, we obtain piecewise $C^{m+1,\gamma}$ regularity for linear elliptic systems with piecewise $C^{m,\gamma}$ coefficients which also come from composite materials.
This answers the open problem suggested by Li and Vogelius in 2000 and Li and Nirenberg in 2003.
Zoom link : https://us02web.zoom.us/j/85151918173?pwd=MCtaTTAzUktZOXVpaFlYdHpBcU5SUT09
2021. 8. 4. (Wed) 16:30-18:00 Room Online
Lukas Schoug (University of Cambridge)

Regularity of the SLE$_4$ uniformizing map and the SLE$_8$ trace

ABSTRACT. The Schramm-Loewner evolution (SLE) is a one-parameter family of random planar fractal curves, which has been of considerable interest since their introduction by Schramm in 1999, as they arise as scaling limits in several two-dimensional statistical mechanics models at criticality. Choosing the parameter $\kappa$ to be either 4 or 8 results in special behaviour, as $\kappa = 4$ ($\kappa = 8$) is the largest (resp. smallest) $\kappa$ such that SLE$_\kappa$ curves are simple (resp. space-filling). As such, regularity results in those cases differ significantly from the cases of other values of $\kappa$. We will discuss recent results on the modulus of continuity of the SLE$_4$ uniformizing map and the SLE$_8$ trace, as well as a byproduct of our analysis, concerning the conformal removability of SLE$_4$. The talk is based on joint work with Konstantinos Kavvadias and Jason Miller.
2021. 8. 3. (Tue) 10:00-11:00 Room Online
Dapeng Zhan (Michigan State University)

Two-curve Green’s function for Schramm-Loewner evolution

ABSTRACT. Schramm-Loewner evolution (SLE$_\kappa$) is a one-parameter family of random fractal curves growing in plane domains. Given an SLE$_\kappa$ curve $\gamma$ with $\kappa\in(0,8)$ in a domain $D$ and a point $z_0$, the Green's function for $\gamma$ at $z_0$ is the limit $G(z_0):= \lim_{r\downarrow 0} r^{-\alpha} \mathbb{P}[\mbox{dist}(z_0,\gamma) < r]$, for some positive exponent $\alpha$, if the limit converges and is not trivial, i.e., not zero. There are several variations. The $z_0$ may be an interior point or a boundary point of $D$. In the former case $\alpha$ is related to the Hausdorff dimension $d$ of $\gamma$ by $\alpha=2-d=1-\frac\kappa 8$. The single point $z_0$ may be replaced by a number of points $z_1,\dots, z_n$.
In this talk, I will describe a method of proving the existence of Green's function for a pair of SLE$_\kappa$ curves $(\gamma_1,\gamma_2)$ in a multiple-SLE$_\kappa$ configuration. The function at a point $z_0$ is the limit $ \lim_{r\downarrow 0} r^{-\alpha} \mathbb{P}[\mbox{dist}(z_0,\gamma_j) < r, j=1,2]$. The $z_0$ could be an interior point or a boundary point. The method can also be used to derive a Green's function related to cut-points of a single SLE curve.
2021. 7. 30. (Fri) 11:00-12:30 Room Online
Jehan Oh (Kyungpook National University)

Regularity results for the nonlinear thin obstacle problem with double phase in the borderline case

ABSTRACT. In this talk, we consider nonlinear thin obstacle problems involving the energy functional with double phase. The energy functional under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm according to the modulating coefficient, which can be regarded as a borderline case of the double phase functional with (p,q)-growth. We provide some conditions on the modulating coefficient and establish regularity results for the thin obstacle problem.
Zoom link : https://us02web.zoom.us/j/85151918173?pwd=MCtaTTAzUktZOXVpaFlYdHpBcU5SUT09
2021. 7. 14. (Wed) 10:00-11:00 & 14:00-15:00 Room Online
Hyeonbae Kang (Inha University)

The spectral theory of the Neumann-Poincare operator: Geometry, Analysis and Applications, I, II

ABSTRACT. The Neumann-Poincare operator (NPO), which is also called the double layer potential, is a boundary integral operator naturally arising when solving classical boundary value problems using layer potentials. Study on the NPO has a long history: it traces back to C. Neumann and Poincare as the name suggests; it gave birth to the Fredholm theory of integral equations and theory of singular integral operators. Lately, interest in the NPO, especially in its spectral properties, has been revived in relation to plasmonic resonances for sub-wavelength imaging. We are witnessing outburst of research on this subject. The purpose of this lecture is to review some (only some) of these developments. Topics to be covered in two talks include: visibility and invisibility via polarization tensors; decay rate of eigenvalues and surface localization of plasmon; singular geometry and essential spectrum; analysis of stress; structure of elastic NPO. We will discuss some problems for future investigation.
Zoom link (10:00-11:00) : https://us02web.zoom.us/j/87219864789
Zoom link (14:00-15:00) : https://us02web.zoom.us/j/84959402016
2021. 7. 8. (Thu) 16:00-17:00 Room 1423
Un Cig Ji (Chungbuk National University)

Convergences of Weighted and Higher Order Cesaro Means in Banach Spaces

ABSTRACT. In this seminar, motivated by the weighted ergodic theorems and infinite dimensional Laplacians (e.g. Levy Laplacian and more extendedly, exotic Laplacian), we discuss the strong convergence of (general) weighted and higher order Cesro means in Banach spaces. As applications, we discuss the weighted and higher order mean ergodic and Birkhoff type ergodic theorems. Also, we discuss a characterization of the ergodicity of measure preserving maps in terms of mixing and higher order mixing properties.
2021. 7. 8. (Thu) 10:00-11:00 & 14:00-15:00 Room Online
Dongho Chae (Chung-Ang University)

Mathematical Challenges in Fluid Mechanics I, II

ABSTRACT. In this lecture we discuss some mathematical problems in the fluid mechanics. Many of the physical systems are described partial differential equations, and the mathematical study of these equations could have important and direct physical significances. In general, in the cases when the equations are nonlinear and nonlocal systems the study of solutions to the equations are much more difficult than the other cases. The fluid equations we are interested in corresponds to these cases. Among others we are particularly interested in the problem of singularities in the incompressible Navier-Stokes and the Euler equations. We emphasize that singularities in the solution of partial differential equations has direct physical importance in many cases. One fashionable example is the singularity of the Einstein field equation, which is nothing but the black hole. The other one is the singularities in the compressible Euler equations, which correspond to the shock waves. The singularities in the Navier-Stokes and the Euler equations are less understood than the two above cases. In the first part of this lecture we introduce, in general terms, the problem of singularities in the fluids. In the second part we focus on some of the recent studies on the problem, related to the scenario of Type I singularities.
Zoom link (10:00-11:00) : https://us02web.zoom.us/j/89261183867
Zoom link (14:00-15:00) : https://us02web.zoom.us/j/84617506761
2021. 6. 18. (Fri) 17:00-18:00 Room Online
Markus Ebke (Bielefeld University)

[GS_M_APP] Symplectic non-Hermitian random matrices - Skew-orthogonal polynomials and universal scaling limits

ABSTRACT. Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. They are characterised by a matrix-valued kernel of skew-orthogonal polynomials, however finding an appropriate set of polynomials for a given matrix ensemble is difficult.
In this talk I will present an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation. Additionally, for the symplectic elliptic Ginibre ensemble I will show how to compute the microscopic large-N limit of the kernel at the origin and prove its universality.
This is based on joint work with Gernot Akemann and Iván Parra. If time permits, I will also mention some results from an ongoing project with Sung-Soo Byun.
2021. 6. 18. (Fri) 16:00-17:00 Room Online
Akansha Sanwal (Bielefeld University)

[GS_M_APP] Local well-posedness for the Zakharov system in dimension $d \leqslant 3$

ABSTRACT. We consider the Zakharov system in dimension $d \leqslant 3$ and show that it is locally well-posed in Sobolev spaces $H^s × H^l$. We construct new solution spaces by modifying the $X^{s,b}$ spaces and use the contraction mapping principle. The result obtained is sharp up to endpoints.
2021. 6. 15. (Tue) 17:00-18:00 Room Online
Simon Brendle (Columbia University)

Classification of ancient solutions to 3D Ricci flow, part II

ABSTRACT. In this lecture, we will discuss the classification of noncompact ancient kappa-solutions in dimension 3. In particular, we show that every noncompact ancient kappa-solution in dimension 3 is rotationally symmetric. The proof uses a continuity argument. It relies, among other things, on the Neck Improvement Theorem from the previous lecture.
Please visit "https://sites.google.com/view/choiks/seminar" for Zoom meeting information.
2021. 6. 11. (Fri) 16:00-17:00 Room Online
Sungsoo Byun (Seoul National University)

Real eigenvalues of elliptic random matrices

ABSTRACT. In this talk, I will discuss the real eigenvalues of the real elliptic Ginibre matrix, the model which provides a natural bridge between Hermitian and non-Hermitian random matrix theories. In the maximally non-Hermitian regime, which corresponds to the matrix model with real i.i.d. Gaussian entries, it was pioneered by Edelman, Kostlan, and Shub that the number of real eigenvalues is of order $\sqrt{N}$, where $N$ is the size of the matrix. Moreover, it can be heuristically conjectured that as a real random matrix becomes more symmetric, it gets more real eigenvalues.
I will demonstrate that such a statement can be made rigorous by presenting the large-N expansion of the mean and the variance of the number of real eigenvalues in the almost-Hermitian regime, where one can observe a non-trivial transition between real i.i.d. and real symmetric random matrices. Furthermore, I will explain the limiting empirical distributions of the real eigenvalues, which interpolate the Wigner semicircle law and the uniform distribution. The proofs are based on the skew-orthogonal representation of the correlation kernel due to Forrester and Nagao.
This is joint work with Nam-Gyu Kang (KIAS), Ji Oon Lee (KAIST) and Jinyeop Lee (LMU München).
2021. 6. 10. (Thu) 17:00-18:00 Room Online
Simon Brendle (Columbia University)

Classification of ancient solutions to 3D Ricci flow, part I

ABSTRACT. It was shown by Perelman that any finite-time singularity of the Ricci flow in dimension 3 is modeled on an ancient kappa-solution. By definition, these are solutions to the Ricci flow which are defined for $t \in (-\infty,T]$ and satisfy certain extra conditions (most importantly, a noncollapsing condition). In dimension 3, there is by now a complete classification of all ancient kappa-solutions. As conjectured by Perelman, the only noncompact examples are the shrinking cylinders (and their quotients), and the rotationally symmetric Bryant soliton. In this lecture, we will discuss some of the main technical ingredients in the proof. This includes the Neck Improvement Theorem, which asserts that a neck tends to become more symmetric under the evolution.
Please visit "https://sites.google.com/view/choiks/seminar" for Zoom meeting information.
2021. 6. 9. (Wed) 10:30-12:00 Room 1423
Hwijae Son (KAIST)

Recent advances in neural network methods for solving PDEs

ABSTRACT. In this talk, I would like to introduce some topics regarding the deep neural network methods for solving PDEs numerically. First, I will first introduce a fundamental concept of physics-informed neural networks and efforts to improve the convergence of physics-informed neural networks. Then I will talk about the deep Ritz method, a recent work by E and Yu (2017), together with the convergence result by Mullier and Zeinhofer (2020).
2021. 6. 8. (Tue) 17:00-18:00 Room Online
Simon Brendle (Columbia University)

The proof of the isoperimetric inequality for minimal surfaces

ABSTRACT. In this lecture, we will discuss a Michael-Simon-type inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Our inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2, thereby answering a question going back to work of Carleman. The proof is inspired by, but does not actually use, optimal transport.
Please visit "https://sites.google.com/view/choiks/seminar" for Zoom meeting information.
2021. 6. 8. (Tue) 16:00-18:00 Room Online
Chuhee Cho (Seoul National University)

The Bochner-Riesz problem: an old approach revisited

ABSTRACT. We consider the Bochner-Riesz operator. We study recent results for the operator obtained by Shaoming Guo et al. [1]. This result is established by using a pseudoconformal transformation and recent techniques developed to study the Fourier restriction problem. In this talk we focus on the Bochner-Riesz problem in $R^3$ and discuss the idea of their approach.
References
[1] Shaoming Guo, Changkeun Oh, Hong Wang, Shukun Wu, and Ruining Zhang. The Bochner-Riesz problem: an old approach revisited, arXiv, 2104.11188
2021. 6. 3. (Thu) 17:00-18:00 Room Online
Simon Brendle (Columbia University)

Singularity models in 3D Ricci flow (KIAS Expositions Lecture)

ABSTRACT. The Ricci flow is a natural evolution equation for Riemannian metrics on a given manifold. From a PDE perspective, the Ricci flow is a system of linear parabolic equations, which can be viewed as the heat equation analogue of the Einstein equations in general relativity. The central problem in the field is to understand singularity formation. In other words, what does the geometry look like at points where the curvature is large? In his spectacular 2002 breakthrough, Perelman achieved a qualitative understanding of singularity formation in dimension 3; this is sufficient for topological conclusions. In this lecture, we will discuss recent developments which have led to a complete classification of all the singularity models in dimension 3.
Zoom link : https://us02web.zoom.us/j/89765910165
2021. 6. 1. (Tue) 17:00-18:00 Room Online
Simon Brendle (Columbia University)

Minimal surfaces and the isoperimetric inequality (KIAS Expositions Lecture)

ABSTRACT. The isoperimetric inequality has a long history, going back to the legend of Queen Dido. In this lecture, I will discuss the isoperimetric inequality and how it relates to the calculus of variations, the notion of mean curvature, and minimal surface theory.
Zoom link : https://us02web.zoom.us/j/81725280484
2021. 5. 28. (Fri) 16:00-18:00 Room 8309
Chuhee Cho (Seoul National University)

A necessary condition for the Schrodinger maximal function

ABSTRACT. In this talk, we consider the maximal estimate for the Schrodinger equation. We investigate necessary conditions for the maximal estimate and provide a rigorous derivation of an example due to Jean Bourgain ([1]).
[1] J. Bourgain, A note on the Schrodinger maximal function, J. Anal. Math. 130 (2016), 393-396.
2021. 5. 25. (Tue) 17:00-18:00 Room Online
Carina Geldhauser (Lund University)

[GS_M_APP] Discrete models for atmospheric turbulence

ABSTRACT. In this talk we introduce a discrete model for atmospheric turbulence, which was originally derived by Helmholtz from the Euler equations. We state some of it basic properties and show how we can derive an effective PDE, the so-called mean field limit, from the discrete Hamiltonian system, by using a variational principle. Furthermore, we discuss the extension of these methods to generalized surface quasigeostrophic models and show how tools from probability theory can help us to get more information about turbulence phenomena. The content of this talk is based on work in collaboration with Marco Romito (Uni Pisa), the presentation tries to avoid technicalities and to be accessible to a mixed audience.
2021. 5. 18. (Tue) 16:00-18:00 Room Online
Hun Hee Lee (Seoul National University)

What is abstract harmonic analysis?

ABSTRACT. 이 강연에서는 abstarct harmonic analysis의 몇 가지 전통적인/최근의 연구주제를 통해 Euclidean harmonic analysis 연구와의 차이점과 유사성에 대해 청중들과 같이 고민해보는 기회를 가져보고자 한다.
https://snu-ac-kr.zoom.us/j/8941592643?pwd=L2gxa3k4V2J1V1dFcTNHUnliRkdpdz09
회의 ID: 894 159 2643 암호: 273190
2021. 5. 13. (Thu) 17:00-18:00 Room Online
Marvin Weidner (Bielefeld university)

[GS_M_APP] Harnack Inequalities for Nonsymmetric Nonlocal Divergence Form Operators

ABSTRACT. In this talk we consider nonlocal equations driven by divergence form operators with a nonsymmetric jumping kernel. Such operators consist of a fractional order diffusion part and a nonlocal drift part of lower order. Our emphasis is on Hölder regularity estimates and (full) Harnack inequalities for solutions to such equations. This is joint work with Moritz Kaßmann (Bielefeld University).
2021. 5. 13. (Thu) 16:00-17:00 Room Online
Marco Rehmeier (Bielefeld university)

[GS_M_APP] Nonuniqueness in law for stochastic hypodissipative Navier-Stokes equations

ABSTRACT. We study the incompressible hypodissipative Navier--Stokes equations with dissipation exponent 0<α<1/2 on the three-dimensional torus perturbed by an additive Wiener noise term and prove the existence of an initial condition for which distinct probabilistic weak solutions exist.
To this end, we employ convex integration methods to construct a pathwise probabilistically strong solution, which violates a pathwise energy inequality up to a suitable stopping time. This paper seems to be the first in which such solutions are constructed via Beltrami waves instead of intermittent jets or flows in a stochastic setting.
2021. 5. 11. (Tue) 16:00-18:00 Room Online
Sanghyuk Lee (Seoul National University)

Maximal estimate for averages over space curves

ABSTRACT. The study of maximal functions has a long tradition in harmonic analysis. The most typical example is the Hardy-Littlewood maximal function whose role in analysis can not be overestimated. The maximal estimates for more singular averages over hypersurfaces have been extensively studied since Stein’s work on the spherical maximal function in the 1970's. However, L^p boundedness of maximal average over submanifolds with codimensions bigger two, as it turns out, is much more involved. No result has been established until recently. In this talk, we consider the maximal operator M which is defined by averages over a smooth space curve with nonvanishing curvature and torsion. We settle the problem of L^p boundedness of M. That is to say, we show that M is bounded on L^p if and only if p>3. The talk is based on recent joint work with Hyerim Ko and Sewook Oh.
( https://snu-ac-kr.zoom.us/j/8941592643?pwd=L2gxa3k4V2J1V1dFcTNHUnliRkdpdz09 )
( ID: 894 159 2643 )
( PW: 273190 )
2021. 4. 13. (Tue) 16:00-18:00 Room Online
Seheon Ham (Seoul National University)

Estimates for the spherical mean and Falconer's distance set problem

ABSTRACT. We study recent progress on Falconer's distance set problem. We begin with Mattila's approach to the problem which is based on spherical average estimates for the Fourier transform of measures. We also study recent improvements due to Guth-Iosevich-Ou-Wang, which made a breakthrough for obstruction raised in Mattila's method. In this talk, we discuss a Sobolev estimate for convolution with the spherical measure which is one of the main ingredients of their work.
2021. 3. 18. (Thu) 17:00-19:00 Room Online
Simone Rademacher (Institute of Science and Technology Austria)

A large deviation principle in many-body quantum dynamics

ABSTRACT. We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates, that are consistent with central limit theorems that have been established in the last years.
2021. 3. 16. (Tue) 17:00-18:00 Room Online
David Garcia-Zelada (Aix-Marseille University)

Radial determinantal Coulomb gases

ABSTRACT. Determinantal Coulomb gases, or random normal matrix models, are particularly simple to study when the potential is radial. I will show that, even in the radial setting, there are interesting limit point process behaviors we may explore. Moreover, I will explain why, in those cases, the regularity assumptions can be reduced to a minimum while the proofs turn out to be fairly straightforward. This talk will be mainly based on arXiv:1812.11170.
2021. 3. 9. (Tue) 14:00-15:00 Room Online
Woocheol Choi (Sungkyunkwan University)

Communication-Computation balanced distributed convex optimization

ABSTRACT. Distributed convex optimization has received a lot of interest from many researchers since it is widely used in various applications, containing wireless network sensor and machine learning. Recently, A. S. Berahas et al (2018) introduced a variant of the distributed gradient descent called the Near DGD+ which combines nested communications and gradient descent steps. They proved that this scheme finds the optimum point using a constant step size when the target function is strongly convex and smooth function. In the first part, we show that the scheme attains O(1/t) convergence rate for convex and smooth function. In addition we obtain a convergence result of the scheme for quasi-strong convex function. In the second part, we use the idea of Near DGD+ to design a variant of the push-sum gradient method on directed graph. This talk is based on a joint work with Doheon Kim and Seok-bae Yun.
Please visit "https://sites.google.com/view/choiks/seminar" for Zoom meeting information.
2021. 3. 8. (Mon) 16:00-18:00 Room Online
Phan Thanh Nam (LMU Munich)

Excitations in the Gross-Pitaevskii regime

ABSTRACT. We study the Bose-Einstein condensates of an inhomogeneous trapped Bose gas in the Gross-Pitaevskii regime. We show that the ground states converge to the Gross-Pitaevskii minimizer and provide with the optimal convergence rate. Our work extends a previous result of Boccato, Brennecke, Cenatiempo and Schlein on the homogeneous gas. Our method relies on the idea of 'completing the square', inspired by recent works of Brietzke, Fournais and Solovej on the Lee-Huang-Yang formula, and an abstract estimate for the quadratic Hamiltonians on Fock space which is of general interest.
2021. 3. 8. (Mon) 14:00-15:00 Room Online
Woocheol Choi (Sungkyunkwan University)

An introduction to the convex optimization

ABSTRACT. In this talk, we review the basic concepts and results for the convex optimization, especially the gradient descent method.
Please visit "https://sites.google.com/view/choiks/seminar" for Zoom meeting information.
2021. 3. 8. (Mon) 11:00-12:00 Room Online
Dongnam Ko (Catholic University of Korea)

Dynamics and control for multi-agent networked systems: a graph limit method

ABSTRACT. We analyze the dynamics of multi-agent collective behavior models and their control theoretical properties. We first consider a finite difference of heat equation and consider its nonlocal interaction dynamics. Then, we address the control problem with the spatial semi-discretization of nonlocal diffusive equations, which related to the graph limit suggested in collective behavior problems. Apart from this existence theory, we also consider numerical methods to construct approximative control functions. Our approach lies in the theory of nonlocal diffusive equations, which is not PDE but a transport-type equations.
2021. 3. 5. (Fri) 16:00-18:00 Room Online
Marcin Napirkowski (University of Warsaw)

Recent advances in the study of the dynamics of Bose gases

ABSTRACT. The last ten years have witnessed a rapid development in the study of dynamics of quantum Bose gases. This led to a rigorous derivation of the non-linear Schrodinger equation in various situations and setups. In my talk I shall review the existing literature, explain the differences between various approaches and point at possible directions of futture research.
2021. 3. 4. (Thu) 16:00-18:00 Room Online
Phan Thanh Nam (LMU Munich)

Derivation of the Gross-Pitaevskii equation for dilute Bose gases

ABSTRACT. We study a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. The derivation of the Bose-Einstein condensation in this case was first proved by Elliott H. Lieb and Robert Seiringer in 2002. In this talk, I will represent an alternative proof, which is based on (1) a combination of Dyson's lemma, (2) the quantum de Finetti theorem, and (3) a second moment estimate for ground states of the effective Dyson Hamiltonian. The proof applies equally well to the case where magnetic fields or rotation are present. This is joint work with Nicolas Rougerie and Robert Seiringer.
2021. 3. 3. (Wed) 16:00-18:00 Room Online
Phan Thanh Nam (LMU Munich)

Bogoliubov theory of the excitation spectrum

ABSTRACT. We study Bogoliubov’s correction to Hartree’s theory of an interacting Bose gas in the mean-field regime. We show that if the Bose-Einstein condensate exists and non-degenerate, then the excited particles can be effectively described by a quadratic Hamiltonian on Fock space which is exactly the quantization of the Hessian of the Hartree energy functional. This leads to a rigorous formula for the excitation spectrum of the interacting Bose gases.
2021. 3. 2. (Tue) 16:00-18:00 Room Online
Phan Thanh Nam (LMU Munich)

Bose-Einstein condensation and mean-field approximation

ABSTRACT. The Bose-Einstein condensation (BEC) is the phenomenon when many quantum particles occupy a common one-body state. Although this was predicted by Bose and Einstein in 1924 based on their computation on the non-interacting gas, the rigorous understanding of the BEC for interacting systems remain a major question in mathematical physics. In this talk, I will discuss a general strategy to derive the BEC for a large class of weakly interacting Bose gases in the mean-field regime.
2021. 2. 25. (Thu) 16:00-18:00 Room Online
Peter Pickl (Institut der Universitat Munchen)

Derivation of the Vlasov equation

ABSTRACT. The rigorous derivation of the Vlasov equation from Newtonian mechanics of N Coulomb-interacting particles is still an open problem.In the talk I will present recent results, where an N-dependent cutoff is used to make the derivation possible. The cutoff is removed as the particle number goes to infinity. Our result holds for typical initial conditions, only. This is, however, not a technical assumption: one can in fact prove deviation from Vlasov dynamics for special initial conditions for the system we consider.
2021. 2. 25. (Thu) 16:00-17:00 Room 1423
Jong-Do Park (Kyung Hee University)

Bergman kernel for complex ellipsoids and hypergeometric functions

ABSTRACT. The Bergman kernel is one of the important tools in several complex variables. In general, it is hard to compute the explicit form of the Bergman kernel for a given bounded domain. In this talk, we explain the computation of the Bergman kernel for some complex ellipsoids. It is expressed in terms of hypergeometric functions which are solutions of some second-order differential equations. We explain the boundary behavior of the Bergman kernel function using the theory of hypergeometric functions. We also explain the recent results on the intersection of cylinder-type domains.
2021. 2. 24. (Wed) 17:00-18:00 Room Online
Sungsoo Byun (Seoul National University)

Almost-Hermitian random matrices and bandlimited point processes: microscopic scaling limits

ABSTRACT. In this talk, I will discuss almost Hermitian random matrices associated with classical Gaussian and Laguerre unitary ensembles.
Based on the approach of Ward’s equation and cross-section convergence, I will explain a way to deduce the bulk scaling limits, which interpolate sine and Ginbre point processes. Moreover I will present the edge scaling limits, which provide non-Hermitian extensions of Airy/Bessel point processes and explain some generalisations of bulk scaling limits.
This is based on joint work with Yacin Ameur.
2021. 2. 23. (Tue) 14:00-15:00 Room Online
Chanho Min (Ajou University)

Introduction to Deep Q Learning and its application to finance

ABSTRACT. This seminar introduces the Deep Q Learning, one of the most popular methods of reinforcement learning and their application to finance problems. Reinforcement learning is a popular model of the learning problems through trial-and-error interactions in a certain given environment. This seminar will discuss the mathematical formulation of Deep Q learning and how each component plays a crucial role in agent learning. Finally it provides real world application in finance, and shows how reinforcement learning can outperform humans even with limited data.
2021. 2. 22 (Mon) 17:00-18:00 Room Online
Sungsoo Byun (Seoul National University)

Almost-Hermitian random matrices and bandlimited point processes: vertical cross-sections and Ward’s equations

ABSTRACT. In this talk, I will discuss a class of bandlimited ensembles in the interface between dimensions 1 and 2, and present certain theorems of a general character.
In particular, I will introduce a property called “cross-section convergence”, which relates the cross-sections of the density of eigenvalues with the equilibrium density for the corresponding Hermitian ensemble. Moreover, combined with the theory of Ward’s equation, I will explain how the cross-section convergence can be used to uniquely determine the bulk scaling limits provided that the latter can be shown to be translation invariant.
This is based on joint work with Yacin Ameur.
2021. 2. 18 (Thu) 17:00-19:00 Room Online
Nicolas Rougerie (Université Grenoble Alpes and CNRS)

Semiclassical limit for almost fermionic anyons

ABSTRACT. In two-dimensional space there are possibilities for quantum statistics continuously interpolating between the bosonic and the fermionic one. Quasi-particles obeying such statistics can be described as ordinary bosons and fermions with magnetic interactions. We study a limit situation where the statistics/magnetic interaction is seen as a "perturbation from the fermionic end". We vindicate a mean-field approximation, proving that the ground state of a gas of anyons is described to leading order by a semi-classical, Vlasov-like, energy functional. The ground state of the latter displays anyonic behavior in its momentum distribution. Our proof is based on coherent states, Husimi functions, the Diaconis-Freedman theorem and a quantitative version of a semi-classical Pauli pinciple.
2021. 2. 18 (Thu) 17:00-18:00 Room Online
Gaultier Lambert (University of Zurich)

Applications of the theory of Gaussian multiplicative chaos to random matrices

ABSTRACT. Log-correlated fields are a class of stochastic processes which describe the fluctuations of some key observables in different probabilistic models in dimension 1 and 2. For instance in random domino tilings, generalized random energy model or the characteristic polynomials of random matrices. Gaussian multiplicative chaos is a renormalization procedure which aims at defining the exponential of a Log-correlated field in the form of a family of random measures. These random measures can be thought of as describing the extreme values of the underlying field. In this talk, I will present some applications of this theory to study the logarithm of the characteristic polynomial of some random matrices. I will focus on the Ginibre ensemble and also mention some results for the Gaussian unitary ensemble and circular beta ensembles.
2021. 2. 18 (Thu) 14:00-16:00 Room 8309
Junha Kim (Chung-Ang University)

Illposedness of the incompressible Euler equations at critical regularity

ABSTRACT.
2021. 2. 17. (Wed) 17:00-18:00 Room Online
Sungsoo Byun (Seoul National University)

A non-Hermitian generalisation of the Marchenko–Pastur distribution: from the circular law to multi-criticality

ABSTRACT. In this talk, I will discuss complex eigenvalues of the product of two rectangular complex Ginibre matrices that are correlated through a non-Hermiticity parameter. I will present the limiting spectral distribution of the model, which provides a non-Hermitian extension of the Marchenko-Pastur distribution. Moreover I will explain the microscopic behaviours of the model in the Dirac picture, which include the limiting correlation kernel at multi-criticality, where the interior of the spectrum splits into two connected components.
This is based on joint work with Gernot Akemann and Nam-Gyu Kang.
2021. 2. 17. (Wed) 16:00-17:00 Room Online
Dongnam Ko (Catholic University of Korea)

Dynamics and control for multi-agent networked systems: a finite difference approach

ABSTRACT. We analyze the dynamics of multi-agent collective behavior models and their control theoretical properties. We first derive a large population limit to parabolic diffusive equations and then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group.
2021. 2. 15. (Mon) 16:00-18:00 Room Online
Widmayer Klaus (École polytechnique fédérale de Lausanne)

On asymptotic stability in the Vlasov-Poisson system

ABSTRACT. In this talk we discuss some recent work on the asymptotic stability of two classic equilibria of the Vlasov-Poisson system: vacuum and a point charge. Starting with an overview of the classic theory in these settings (which guarantees the global existence of smooth solutions under suitable conditions on the initial data) we introduce a new approach that combines a Lagrangian analysis of the linearized problem with an Eulerian PDE framework in the nonlinear analysis, all the while respecting the symplectic structure. For sufficiently small data, in both of the above cases the asymptotic behavior is then seen to be a modified scattering dynamic, which we can precisely quantify.
This is joint work with Benoit Pausader (Brown University).
2021. 2. 5. (Fri) 16:00-17:00 Room 1423
Hyeong-Ohk Bae (Ajou University)

Interaction of Particles and an Incompressible Fluid

ABSTRACT. We present a new coupled kinetic-fluid model for the interactions between Cucker-Smale(C-S) flocking particles and incompressible fluid on the periodic spatial domain. Our coupled system consists of the kinetic Cucker-Smale equation and the incompressible Navier-Stokes equations, and these two systems are coupled through the drag force. For the proposed model, we provide a global existence of weak solutions and a priori time-asymptotic exponential flocking estimates for any smooth flow, when the kinematic viscosity of the fluid is sufficiently large. The velocity of an individual C-S particles and fluid velocity tend to the averaged time-dependent particle velocities exponentially fast.
2021. 2. 5. (Fri) 15:00-17:00 Room Online
Younghun Hong (Chung-Ang University)

Long-time asymptotic behavior for the Vlasov-Poisson system

ABSTRACT. The Vlasov-Poisson system is a kinetic equation describing a large number of classical particles interacting by the Newtonian/Coulombian force. For small data solutions, Bardos and Degond obtained a long-time decay estimate for density functions in 1985, and then S. Choi and S. Kwon proved modified scattering. Very recently, this problem has been revisited by Ionescu, Pausader, Wang and Widmayer using dispersive PDE techniques motivated by the resonance analysis, and they provided an alternative approach. In this talk, we too revisit the problem but using a different toolbox, namely Strichartz estimates for almost free transport flow. This talk is based on joint work with Jin Woo Jang (Bonn) and Maja Taskovic (Emory).
2021. 2. 5. (Fri) 14:00-15:00 Room Online
Jinmyoung Seok (Kyonggi University)

Uniqueness of McCann's binary star.

ABSTRACT. Since a ground breaking work by Cazenave-Lions in 1982, showing uniqueness (up to symmetries) of variationally constructed solutions to Hamiltonian PDEs has played an indispensable role for verifying their orbital stability. In this talk, we discuss how to obtain the uniqueness of a family of binary star solutions to the Euler-Poisson equations, variationally constructed by McCann in 2006. Main methodology is based on perturbation arguments crucially relying on the exact asymptotic behaviors of solutions.
2021. 2. 5. (Fri) 10:00-14:00 Room 8406
Junha Kim (Chung-Ang University)

Asymptotic behavior of stratified solutions for the 3D Boussinesq equations with a velocity damping term

ABSTRACT.
2021. 1. 29. (Fri) 14:00-16:00 Room 8309
Junha Kim (Chung-Ang University)

Illposedness of active scalars at critical regularity II

ABSTRACT.
2021. 1. 29. (Fri) 10:00-12:00 Room 8309
In-Jee Jeong (KIAS)

Illposedness of active scalars at critical regularity I

ABSTRACT.
2021. 1. 28. (Thu) 14:00-16:30 Room Online
Doyoon Kim (Korea University)

Parabolic Sobolev embeddings and parabolic equations with unbounded lower-order coefficients.

ABSTRACT. We discuss Sobolev spaces for parabolic equations. In particular, we introduce refined Sobolev embeddings for solutions to parabolic equations in divergence form. Using these results, we prove the unique solvability in Sobolev spaces of parabolic equations when the lower-order coefficients are in appropriate mixed $L_{p,q}$ spaces (not necessarily bounded).
This is joint work with Seungjin Ryu and Kwan Woo.
2021. 1. 26. (Tue) 15:00-16:00 Room Online
Bongsuk Kwon (UNIST)

Euler-Poisson system and plasma phenomena

ABSTRACT. We consider the Euler-Poisson system describing dynamics of ions in electrostatic plasmas and related physical phenomena. These include plasma sheaths, solitons and plasma shock waves. We discuss the related mathematical issues and open problems.
2021. 1. 26. (Tue) 14:00-15:00 Room Online
Young-Pil Choi (Yonsei University)

Quantified overdamped limit for Vlasov-Fokker-Planck equations with singular interaction forces

ABSTRACT. In this talk, I will discuss a quantified overdamped limit for kinetic Vlasov-Fokker-Planck equations with nonlocal interaction forces. We provide explicit bounds on the error between solutions of that kinetic equation and the limiting equation, which is known under the names of aggregation-diffusion equation or McKean-Vlasov equation. Our strategy only requires weak integrability of the interaction potentials, thus in particular it includes the quantified overdamped limit of the kinetic Vlasov-Poisson-Fokker-Planck system to the aggregation-diffusion equation with either repulsive electrostatic or attractive gravitational interactions.
2021. 1. 25. (Mon) 14:00-16:00 Room Online
Seunghyeok Kim (Hanyang University)

Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

ABSTRACT. Under the validity of the positive mass theorem, the Yamabe flow on a smooth compact Riemannian manifold $M$ of dimension $N ≥ 3$ is known to exist for all time $t$ and converges to a metric of constant scalar curvature as $t \to \infty$. We show that if a suitable perturbation, which can be made arbitrarily small and smooth, is imposed on the linear term of the Yamabe flow on $M$, the resulting flow blow-ups at the infinite time, forming singularities each of which looks like a solution of the Yamabe problem on the unit sphere $\mathbb{S}^N$. This is joint work with Monica Musso (University of Bath, UK)
2021. 1. 25. (Mon) 10:00-12:00 Room Online
Seunghyeok Kim (Hanyang University)

Multiple blowing-up solutions to critical elliptic systems in bounded domains

ABSTRACT. In this talk, we consider the existence of families of blowing-up solutions to elliptic systems on smooth bounded domains in the Euclidean space, which are variants of the critical Lane-Emden system and analogous to the Brezis-Nirenberg problem. We see that the governing function of blowing-up points and rates reflects the strong nonlinear characteristic of the system. If time permits, we also discuss how the new ideas and arguments developed here can be used to analyze blowing-up phenomena in related Hamiltonian-type systems. This is joint work with A. Pistoia (Sapienza University of Rome).
2021. 1. 15. (Fri) 14:00-16:00 Room Online
Hyunwoo Kwon (Republic of Korea Air Force Academy/Sogang University)

$W^{1,2+\epsilon}$-results for elliptic equations with drifts in weak $L^n$-spaces

ABSTRACT. We consider the following Dirichlet problems for elliptic equations with singular drift $\mathbf{b}$: \[ \text{(a) } -\operatorname{div}(A \nabla u)+\operatorname{div}(u\mathbf{b})=f,\quad \text{(b) } -\operatorname{div}(A^T \nabla v)-\mathbf{b} \cdot \nabla v =g \quad \text{in } \Omega, \] where $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^n$, $n\geq 2$. Assuming that $\mathbf{b}\in L^{n,\infty}(\Omega)^n$ has non-negative weak divergence in $\Omega$, we establish existence and uniqueness of weak solution in $W^{1,2+\varepsilon}_0(\Omega)$ of the problem (b) when $A$ is bounded and uniformly elliptic. As an application, we prove unique solvability of weak solution $u$ in $\bigcap_{q<2} W^{1,q}_0(\Omega)$ for the problem (a) for every $f\in \bigcap_{q<2} W^{-1,q}(\Omega)$.
2021. 1. 15. (Fri) 10:00-12:00 Room Online
Siming He (Duke University)

Fluid Flows, Chemotaxis and Reactions

ABSTRACT. In this talk, I will present some results on suppressing chemotactic blow-ups and enhancing chemical reactions through passive fluid flows. Enhanced dissipation effect of mixing flows and shear flows play significant roles. I will also discuss a new coupled Chemotaxis-Fluid system (Patlak-Keller-Segel-Navier-Stokes) and the corresponding optimal critical mass result.
2021. 1. 11. (Mon) 16:30-18:00 Room Online
Sanghyuk Lee (Seoul National University)

Almost everywhere convergence of Bochner-Riesz means for the Hermite expansion

ABSTRACT. In this talk we consider almost everywhere convergence of the Bochner-Riesz means of the Hermite expansion and characterize the summability index for which the means converges almost everywhere. For $f\in L^p$ with $p\ge 2$ we show the Bochner-Riesz means converges almost everywhere if the summability index is bigger than $\alpha(p)=\max(n(1/4-1/2p)-1/4, 0)$ and the convergence generally fails if the index is smaller than $\alpha(p)$. This is in striking contrast with the classical Bochner-Riesz means of Fourier transform and Fourier series in that the required summability index is only half of the index for almost everywhere convergence of the classical Bochner-Riesz means of Fourier transform and Fourier series.
2021. 1. 11. (Mon) 15:00-16:30 Room Online
Sanghyuk Lee (Seoul National University)

Sharp $L^p-L^q$ estimates for the Hermite spectral projection

ABSTRACT. This talk primarily concerns the sharp bound on the spectral projection of the Hermite operator in the $L^p$ spaces. In comparison with the spectral projection of the Laplacian, the sharp bound is not so well understood. We consider the estimate for the spectral Hermite projection in general $L^p-L^q$ framework and obtain various new sharp estimates in an extended range. Especially, we provide a complete characterization of the local estimate and prove the endpoint $L^2$--$L^{2(d+3)/(d+1)}$ estimate which has been left open since the work of Koch and Tataru. We also discuss application of the projection estimate to related problems, such as the resolvent estimate for the Hermite operator and Carleman estimate for the heat operator.
2021. 1. 6. (Wed) 15:00-17:00 Room Online
Hantaek Bae (UNIST)

Gevrey Regularity for Some Fluid Equations

ABSTRACT. In this talk, we establish Gevrey regularity of solutions to several fluid equations.
First of all, we show analyticity of solutions to the Navier-Stokes equations (incompressible, inhomogeneous incompressible, compressible) via modified-Gevrey regularity approach.
We then take some models derived from the equations of water-waves. We first deal with Kakutani-Matsuuchi model and explain how to derive decay rates of solutions by establishing Gevrey regularity of order $s<1$. We next prove singularity formation of Green-Naghdi model after showing analyticity of solutions. We finally discuss how to prove ill-posedness of Ambose-Bona-Nicholls model.
2021. 1. 4. (Mon) 14:00-16:00 Room 8309
Junha Kim (Chung-Ang University)

Illposedness of the inviscid SQG equation at critical regularity (part 2)

ABSTRACT.
2021. 1. 4. (Mon) 10:00-12:00 Room 8309
Junha Kim (Chung-Ang University)

Illposedness of the inviscid SQG equation at critical regularity (part 1)

ABSTRACT.

Past Seminars

2020 Seminars
2019 Seminars

ORGANIZERS Sung-Soo Byun - Korea Institute for Advanced Study
Eunhee Jeong - Jeonbuk National University
In-Jee Jeong - Seoul National University
Nam-Gyu Kang - Korea Institute for Advanced Study
Doheon Kim - Korea Institute for Advanced Study
Yehyun Kwon - Korea Institute for Advanced Study
Jaehun Lee - Korea Institute for Advanced Study
Jinyeop Lee - Ludwig-Maximilians-Universität München
Sung-Jin Oh - UC Berkeley & KIAS
Bae Jun Park - Korea Institute for Advanced Study
Jung-Tae Park - Korea Institute for Advanced Study
Sung Chul Park - Korea Institute for Advanced Study
Seong-Mi Seo - KAIST

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ORGANIZERS	Sung-Soo Byun - Korea Institute for Advanced Study
	Eunhee Jeong - Jeonbuk National University
	In-Jee Jeong - Seoul National University
	Nam-Gyu Kang - Korea Institute for Advanced Study
	Doheon Kim - Korea Institute for Advanced Study
	Yehyun Kwon - Korea Institute for Advanced Study
	Jaehun Lee - Korea Institute for Advanced Study
	Jinyeop Lee - Ludwig-Maximilians-Universität München
	Sung-Jin Oh - UC Berkeley & KIAS
	Bae Jun Park - Korea Institute for Advanced Study
	Jung-Tae Park - Korea Institute for Advanced Study
	Sung Chul Park - Korea Institute for Advanced Study
	Seong-Mi Seo - KAIST
	How to visit KIAS
Directions to KIAS	English - Korean