Analysis, PDE & Probability Seminar

Analysis, PDE & Probability Seminar

2024. 11. 15. (Fri) 11:00-12:00 Room 1424
Sehyun Ji (University of Chicago)

Dissipation estimates of the Fisher information for the Landau equation with the Coulomb potential

ABSTRACT. About a year ago, Nestor Guillen and Luis Silvestre proved a significant result, showing the Fisher information is monotone decreasing for the Landau equation with the Coulomb potential. As a consequence, they deduced the solutions do not blow up for $C^1$ initial data. We establish an a priori estimate for the dissipation of the Fisher information. As an application, we show the global existence of smooth solutions for rough initial data in $L^1_5 \cap L\log L$. In this talk, we will begin with an overview of Guillen and Silvestre's proof.
2024. 11. 8. (Fri) 09:00-10:00 Room Online
Yangrui Xiang (University of Nottingham)

Entropy method for mean-field interacting particle systems

ABSTRACT. We study two models: The noisy SIS (Susceptible-Infected-Susceptible) model and a generalized contact process. We investigate the convergence towards equilibrium of the noisy SIS model, which is a stochastic perturbation of SIS Model. In the noisy SIS model, individuals can change states randomly due to external noise. We employ the stochastic Curie-Weiss model as our approach. We find the profile of convergence, which is given as the total variation distance between the marginals of a suitable Gaussian process and its stationary state. This analysis demonstrates that the mixing time is of order O(log n) with the constant determined by the parameters of the model, and the model exhibits cut-off with the window of size O(1). Besides, we derive a quantitative version of the hydrodynamic limit for an interacting particle system inspired by integrate-and-fire neuron models. More precisely, we show that the L2-speed of convergence of the empirical density of states in a generalized contact process defined over a d-dimensional torus of size n is of the optimal order O(nd/2). In addition, we show that the typical fluctuations around the aforementioned hydrodynamic limit are Gaussian, and governed by an inhomogeneous stochastic linear equation.
2024. 11. 5. (Tue) 14:00-15:30 Room 1424
Sungsoo Byun (Seoul National University)

Integrable structure and free energy of Coulomb gases on the torus

ABSTRACT. In this talk, I will discuss Coulomb gases on compact Riemann surfaces. A key topic in Coulomb gas theory is the free energy expansion as the system size tends to infinity. This asymptotic expansion plays a crucial role in establishing the law of large numbers and central limit theorems, and it also finds applications in several large deviation probabilities. Additionally, the coefficients in this expansion encapsulate potential-theoretic, topological, and conformal geometric aspects of the model. I will present some integrable structures of Coulomb gases that allow for the application of the theory of (elliptic) orthogonal polynomials in the genus zero and one cases.
2024. 10. 30. (Wed) 17:00-18:00 Room Online
Zhengyan Wu (Bielefeld University)

Higher Order Fluctuation Expansions for Nonlinear Stochastic Heat Equations in Singular Limits

ABSTRACT. Higher order fluctuation expansions for stochastic heat equations (SHE) with nonlinear, non-conservative and conservative noise are obtained. These Edgeworth-type expansions describe the asymptotic behavior of solutions in suitable joint scaling regimes of small noise intensity and diverging singularity. The results include both the case of the SHE with regular and irregular diffusion coefficients. In particular, this includes the correlated Dawson-Watanabe and Dean-Kawasaki SPDEs, as well as SPDEs corresponding to the Fleming-Viot and symmetric simple exclusion processes. This is joint work with Benjamin Gess and Rangrang Zhang.
2024. 10. 14. (Mon) 16:00-17:00 Room Online
Hayate Suda (Tokyo Institute of Technology)

Scaling limits of a tagged soliton in the randomized box-ball system

ABSTRACT. The box-ball system (BBS) is a cellular automaton that exhibits the solitonic behavior. In recent years, with the rapid progress in the study of the hydrodynamics of integrable systems, there has been a growing interest in BBS with random initial distribution. In this talk, we consider the scaling limits for a tagged soliton in the BBS starting from certain stationary distribution.
2024. 9. 30. (Mon) 10:00-11:00 Room 1423
Jonguk Yang (University of Zurich)

Self-Similarity of 2D Dynamics at the Boundary of Chaos

ABSTRACT. It has been observed in various settings (such as a population of rabbits, thermal convection in a fluid, etc) that the transition from regular dynamics to chaos happens through a sequence of period-doubling bifurcations. Moreover, the quantitative characteristics of this phenomenon appear to be universal, meaning they do not depend on the specific details of the systems under consideration. For 1D unimodal maps, these observations have been rigorously justified through the work of Sullivan, McMullen, and Lyubich using renormalization techniques. Together with S. Crovisier, M. Lyubich, and E. Pujals, we extended this theory to the 2D setting of Hénon-like maps. In this talk, I will provide an overview of our key results and discuss how the tools we developed contribute to the broader study of 2D dynamical systems.
2024. 9. 27. (Fri) 10:00-11:00 Room 1423
Jaehun Lee (KAIST SAARC)

Spectral heat contents for Levy processes

ABSTRACT. For a process $X$ and domain $D$, the spectral heat content $Q^X_D(t):=\int_DP_x(\tau^X_D>t)dx$ represents the total heat remains in $D$ at time $t>0$ with Dirichlet boundary condition. It is known that when $X$ is an isotropic $\alpha$-stable process with $0<\alpha<2$, the asymtotic behavior of $|D|−Q^X_D(t)$ is given by $$|D|−Q^X_D(t)=\begin{cases}O(t^{1/\alpha}),& \alpha\in(1,2),\\O(t),& \alpha\in(0,1).\end{cases}$$ In this talk, we obtain a sharp sufficient condition for the first case $|D|−Q^X_D(t)=O(t^{1/\alpha})$ in the domain with inner and outer ball conditions when $X$ is Levy process. In particular, the result covers both non-isotropic Levy processes and 1-stable processes. This talk is based on the joint work with Hyunchul Park in SUNY.
2024. 9. 20. (Fri) 16:00-17:00 Room Online
Yoël Perreau (University of Tartu)

Diametral diameter 2 properties for subspaces of $L_1$

ABSTRACT. We show that all the diametral diameter two properties are equivalent to the Daugavet property for subspaces of $L_1$. On the way, we also provide some results about the localization of these properties in this setting. We show that reflexive subspaces of $L_1$ contain no diametral points, and, if time permits, discuss special kind of subspaces in which Daugavet and Delta points coincide.
2024. 9. 19. (Thu) 10:00-11:00 Room 1423
Sang-Jun Park (University of Toulouse)

Random matrix approach to quantum entanglement

ABSTRACT. Entanglement of quantum states is one of the most fundamental concepts in quantum information theory, due to its usefulness in various quantum protocols and computations. In this talk, we introduce a mathematical approach to quantum entanglement using the probabilistic method, mainly on random matrix theory, combined with high-dimensional convex analysis. In particular, we will discuss the threshold phenomena in quantum entanglement: the generic entanglement behavior of random quantum states "flips" to its opposite when an intermediate parameter crosses a critical threshold value. Finally, we will present our recent work with Ion Nechita, which describes threshold phenomena in random quantum states with group symmetries.
2024. 9. 11. (Wed) 13:30-14:30 Room 1423
Inwon Kim (UCLA)

The supercooled Stefan problem

ABSTRACT. We will discuss the supercooled Stefan problem, which is a classical model describing the freezing of supercooled water (namely the unstable state of water below freezing temperature). I will introduce the PDE formulation of the problem introduced by Stefan, with some literature that exhibits instability of solutions. I will then describe the variational approach for the problem, which is motivated by the underlying interacting particle systems. As a consequence of our approach we will in particular obtain a global-time existence of solutions in general geometry. The results I will discuss includes joint work with Young-Heon Kim (UBC) and with Sunhi Choi (Arizona).
2024. 9. 6. (Fri) 10:00-11:00 Room 1423
Juyoung Lee (KIAS)

Maximal Functions in Harmonic Analysis

ABSTRACT. Since Stein's seminal work on the spherical maximal function, maximal functions have been extensively studied over the past few decades. The boundedness of these maximal functions in Lebesgue spaces has significant implications in geometric measure theory and is closely related to local smoothing estimates, a central topic in harmonic analysis. In this talk, we will focus on maximal averages over dilated submanifolds in the Euclidean space and in the Heisenberg group. We will provide a brief overview of some landmark results and then discuss recent progress in the study of these maximal functions.
2024. 8. 28. (Wed) 16:00-17:00 Room Online
Van Hao Can (Vietnam Academy of Science and Technology)

Ising model on random graphs

ABSTRACT. While the Ising model is a well-known framework for studying phase transitions in ferromagnetism and cooperative behavior in social networks, random graphs are considered an efficient tool for modeling large complex networks. In this talk, I will review the recent developments in the topic of the Ising model on random graphs, including phase transitions and asymptotic behavior, and mixing time of corresponding Glauber dynamics. This talk is based on joint works with Cristian Giardina, Claudio Giberti, Remco van der Hofstad and Takashi Kumagai.
2024. 8. 20. (Tue) 16:00-17:00 Room 1424
Kihoon Seong (Cornell University)

Gaussian fluctuation of Euclidean $\Phi^4$ QFT

ABSTRACT. I will discuss the asymptotic expansions of the Euclidean $\Phi^4$-measure in the low-temperature regime. Consequently, we derive limit theorems, specifically the law of large numbers and the central limit theorem for the $\Phi^4$-measure in the low-temperature limit. In the second part of the talk, I will focus on the infinite volume limit of the focusing $Phi^4$-measure. Specifically, with appropriate scaling, the focusing $\Phi^4$-measure exhibits Gaussian fluctuations around a scaled solitary wave, that is, the central limit theorem. This talk is based on joint works with Benjamin Gess, Pavlos Tsatsoulis, and Philippe Sosoe.
2024. 7. 22. (Mon) 10:00-11:00 Room 1423
Joonhyun La (KIAS HCMC)

Wave turbulence and some well-posedness results

ABSTRACT. Zoom link: https://us06web.zoom.us/j/81799178907 Wave turbulence refers to the statistical theory of weakly nonlinear dispersive waves. In the weakly turbulent regime of a system of dispersive waves, its statistics can be described via a coarse-grained dynamics, governed by the kinetic wave equation. Remarkably, kinetic wave equations admit exact power-law solutions, called Kolmogorov-Zakharov spectra, which resemble Kolmogorov spectrum of hydrodynamic turbulence, and is often interpreted as a transient equilibrium between excitation and dissipation. In this talk, we will outline a local well-posedness result for kinetic wave equation for a toy model for wave turbulence. The result includes well-posedness near K-Z spectra, and demonstrates a surprising smoothing effect of the kinetic wave equation. The talk is based on the joint work with Pierre Germain (ICL) and Katherine Zhiyuan Zhang (Northeastern).
2024. 7. 1. (Mon) 16:00-17:00 Room 7323
Seung-Yeon Ryoo (California Institute of Technology)

Lipschitz Padded Nested Random Partitions

ABSTRACT. We introduce the concept of a "Lipschitz Padded Nested Random Partition," which is a multiscale version of padded decompositions. Often in geometry, when one wishes to construct an object globally on a geometric object such as a manifold or a discrete metric space, one first constructs it locally and pastes it via some partition of unity; control on the partition of unity leads to control on the resulting global object. We show that the existence of a Lipschitz Padded Nested Random Partition on a doubling metric space implies a sharp version of the Assouad embedding theorem, and that the Heisenberg group admits a Lipschitz Padded Nested Random Partition. This gives an alternative construction of the sharp Assouad embedding for the Heisenberg group. This is a forthcoming joint work with Alan Chang.
2024. 6. 20. (Thu) 16:20-17:00 Room 204 (Soorim)
Kyuhyeon Choi (Seoul National University)

A $\Gamma$-convergence of level-two large deviation for metastable systems: The case of zero-range processes

ABSTRACT. In this talk, we discuss the relationship between the precise asymptotics of the level-two large deviation rate function and the behavior of metastable stochastic systems. It is conjectured that the metastable and pre-metastable behaviors are encapsulated in the level-two large deviation of the markov process of the stochastic system. We rigorously prove this statement for a certain class of zero-range process by developing a methodology for $\Gamma$-convergence in the pre-metastable time scale and linking the resolvent approach to metastability (Landim et al., J Eur Math Soc, 2023. arXiv:2102.00998) with the $\Gamma$-expansion in the metastable time scale.
2024. 6. 20. (Thu) 15:40-16:20 Room 204 (Soorim)
Younghun Jo (Seoul National University)

The Eyring–Kramers law for extinction time of contact process on stars

ABSTRACT. In this talk, we present a precise estimate for the mean extinction time of the contact process with a fixed infection rate on a star graph with $N$ leaves. Specifically, we determine not only the exponential main factor but also the sharp sub-exponential prefactor of the asymptotic formula for the mean extinction time as $N\to\infty$, a level of detail previously available only for complete graphs. We achieved these results by estimating the quasi-stationary distribution on non-extinction using special function theory and refined Laplace's method, and by applying the potential theoretic approach to metastability of non-reversible Markov processes. This novel integration of methodologies provides new insights into the study of the contact process.
2024. 6. 20. (Thu) 15:00-15:40 Room 204 (Soorim)
Seongjae Park (Seoul National University)

Large gap probabilities of complex and symplectic spherical ensembles with point charges

ABSTRACT. The calculation of the asymptotics of the probability that there are no particles in a certain gap, known as the gap probability, is an important problem in point processes. In this talk, I will present the asymptotic expansion of the gap probabilities of complex and symplectic induced spherical ensembles, which can be realized as determinantal and Pfaffian 2D Coulomb gases on the Riemann sphere with the insertion of point charges. More precisely, when the gap is a spherical cap around the poles, we show that the gap probability has an asymptotic behavior of the form \[ \exp(C_1n^2 + C_2n\log n + C_3n + C_4\sqrt{n} + C_5\log n+ C_6 + o(1)). \] Our proof relies on the uniform asymptotics of the incomplete beta function. This is based on joint work with Sung-Soo Byun.
2024. 6. 17. (Mon) 16:00-17:00 Room 1423
Soobin Cho (University of Illinois Urbana-Champaign)

Robust estimates for elliptic nonlocal operators on metric measure spaces

ABSTRACT. In this presentation, I'll explore the analysis of non-local regular Dirichlet forms on metric measure spaces. Our approach relies on three key assumptions: the presence of a strongly local Dirichlet form with sub-Gaussian heat kernel estimates, a tail estimate governing the jump measure outside balls, and a local energy comparability condition. Our primary objectives include establishing function inequalities such as localized Poincaré, cutoff Sobolev, and Faber-Krahn inequalities, highlighting their inherent stability structures and their implications for the regularity of corresponding harmonic functions. Our results are robust in the sense that the constants in estimates remain bounded, provided that the order of the scale function appearing in the tail estimate and local energy comparability condition, maintains a certain distance from zero.
2024. 6. 14. (Fri) 15:00-17:00 Room 8101
Ilwoo Cho (St. Ambrose university)

A Mathematical Model for Functional Extreme Learning Machines: Operator-Algebraic and Free-Probabilistic Approaches

ABSTRACT. In this talk, we consider a mathematical model for functional extreme learning machines (FELMs) in terms of known graph theory, groupoid theory, representation theory, operator theory, operator algebra and free probability theory. These analytic techniques and tools allow us to analyze the constituents of a FELM with respect to operator theory, and access them in terms of free-distributional data.
2024. 6. 13. (Thu) 16:30-17:30 Room Online
Marvin Weidner (University of Barcelona)

Pointwise estimates for parabolic nonlocal equations

ABSTRACT. The celebrated De Giorgi-Nash-Moser theory establishes Hölder regularity of solutions to second order equations in divergence form without any regularity assumptions on the coefficients. This result was a key ingredient in the resolution of Hilbert's XIXth problem in the 1950s. An important contribution to the theory was made by Moser who established a Harnack inequality for solutions to such equations. During the last 20 years, there was a huge interest in the study of similar regularity results for solutions to nonlocal equations governed by operators that are modeled upon the fractional Laplacian. In this talk, we give an overview of this direction of research, focusing on the main difficulties arising from the long range interactions, that lie in the nature of nonlocal models. Moreover, we explain our recent results on the nonlocal parabolic Harnack inequality and Hölder regularity estimates, which complete the regularity program for linear nonlocal equations with bounded measurable coefficients. This talk is based on a joint work with Moritz Kaßmann.
2024. 6. 7. (Fri) 15:30-17:00 Room 8309
Jasang Yoon (University of Texas RGV)

Square root problems of operators II

ABSTRACT. In this second talk we sharpen the statement of a previous result on Square Root Problem for measures and extend its applicability via a new technique that uses the standard inequality of real numbers to generate a diagram of a partial order on the support of a probability measure. Then, we expand the relation of the subnormality of the Aluthge transform of a unilateral weighted shift to the joint subnormality of the spherical and toral Aluthge transforms of a 2-variable weighted shift.
2024. 6. 7. (Fri) 10:00-11:00 Room 1423
Thomas Alberts (University of Utah)

Conformal Field Theory on Multiply Connected Domains

ABSTRACT. We implement a version of conformal field theory on a multiply connected domain. We show that all correlation functions for the Dirichlet Gaussian Free Field are martingale observables for a stochastic Loewner evolution on the domain. As part of it we derive the generalized Eguchi-Ooguri equations, which give an explicit form of Ward's equations that describes the insertion of a stress tensor in terms of Lie derivatives and differential operators depending on the Teichmuller modular parameters. Joint work with Sung-Soo Byun and Nam-Gyu Kang.
2024. 5. 30. (Thu) 17:00-18:00 Room Online
Christian Cobollo (Universitat Politècnica de València)

Linear Dynamics and disjointness in Lipschitz-free spaces

ABSTRACT. There is a recent line of research devoted to studying whether the functor that relates a Lipschitz map $f$ on a metric space $M$ to its corresponding linearization operator $T_f$ on the Lipschitz-free space $\mathcal{F}(M)$ carries information about a given property. Along this talk, we will present some results regarding the study of Linear Dynamics, providing conditions to obtain properties like Hypercyclicity, Disjoint-transitiveness, and the Operator Specification Property in the Lipscthiz-free operators $T_f$ depending on the underlying (non-linear) dynamical system $(M,f)$. Roughly speaking, we will obtain that, provided that a dynamical property has "high enough frequency'', then it passes to the Lipschtiz-free space. This talk is based on a joint work with Alfred Peris.
2024. 5. 24. (Fri) 15:30-17:00 Room 8309
Jasang Yoon (University of Texas RGV)

Square root problems of operators I

ABSTRACT. There are two notions of Square Root Problems: one is for probability measures and the other is for operators. In the first talk we consider the following Square Root Problem for probability measures: Given a positive probability Borel measure $\mu$ (supported on an interval $[a,b] \subseteq \mathbb{R}_+$), does there exist a positive Borel measure $nu$ such that $\mu = \nu * \nu$ holds? (Here $*$ denotes the multiplicative convolution, properly defined on $\mathbb{R}_+$.) This problem is closely connected to the subnormality of the Aluthge transform of a unilateral weighted shift. We develop a criterion to test whether a measure µ admits a square root, and we provide a concrete solution for the case of a finitely atomic measure having at most six atoms.
2024. 5. 23. (Thu) 16:00-17:00 Room Online
Shuta Nakajima (Meiji University)

Upper tail large deviation for the one-dimensional frog model

ABSTRACT. In this talk, we study the upper tail large deviation for the one-dimensional frog model. In this model, sleeping and active frogs are assigned to vertices on $\mathbb{Z}$. While sleeping frogs do not move, the active ones move as independent simple random walks and activate any sleeping frogs. The main object of interest in this model is the asymptotic behavior of the first passage time $T(0,n)$, which is the time needed to activate the frog at the vertex $n$, assuming there is only one active frog at $0$ at the beginning. While the law of large numbers and central limit theorems have been well established, the intricacies of large deviations remain elusive. Using renewal theory, Bérard and Ramírez have pointed out a slowdown phenomenon where the probability that the first passage time $T(0,n)$ is significantly larger than its expectation decays sub-exponentially and lies between $\exp(-n^{1/2 + o(1)})$ and $\exp(-n^{1/3 + o(1)})$. In this article, using a novel covering process approach, we confirm that $1/2$ is the correct exponent, i.e., the rate of upper large deviations is given by $n^{1/2}$. Moreover, we obtain an explicit rate function that is characterized by properties of Brownian motion and is strictly concave. This talk is based on joint work with Van Hao Can, Naoki Kubota.
2024. 5. 17. (Fri) 10:00-11:00 Room 1423
Haakan Hedenmalm (KTH)

Hyperbolic Fourier series and the Klein-Gordon equation

ABSTRACT. We introduce the concept of a hyperbolic Fourier series and apply the main representation theorem to the solutions of the Klein-Gordon equation in 1+1 dimensions.
2024. 5. 9. (Thu) 17:00-18:00 Room Online
Andre Ostrak (University of Agder)

Diameter two properties in spaces of Lipschitz functions

ABSTRACT. Properties, which imply that all specific subsets (e.g. slices) of the unit ball of a Banach space have diameter 2, are referred to as diameter two properties. We give a brief overview of the study of such properties in spaces of Lipschitz functions. Our focus lies on recent results presented in two preprints, showing that the SSD2P, SD2P, and D2P are three different properties in spaces of Lipschitz functions and that the diameter two properties (including LD2P) are distinct from their weak-star counterparts in this class of Banach spaces. Furthermore, we discuss different sufficient conditions for the space of Lipschitz functions to have the SSD2P and SD2P. The talk is mainly based on the results of the following preprints: https://arxiv.org/abs/2205.13287 https://arxiv.org/abs/2404.11430
2024. 5. 3. (Fri) 16:00-17:10 Room 1424
Ki-Ahm Lee (Seoul National University)

Capacity theory for nonlinear equations and its application

ABSTRACT. In this talk we would like to discuss how capacity theory for nonlinear operators shows up in applications: Homogenization for the highly oscillating obstacles and Wiener Criterion for the boundary value problems. At Homogenization theory, we will consider the influence of the capacity of periodically or randomly oscillating obstacles for various nonlinear operators to find the effective equation. For Wiener Criterion for the boundary value problems, we will measure the influence of the capacity of the complement of the domains in shrinking local regions to Weak Harnack Inequalities.
2024. 5. 3. (Fri) 10:00-11:00 Room 1424
Manan Bhatia (MIT)

Mated trees and a Peano curve in the directed landscape

ABSTRACT. A recurrent theme encountered in many models of random geometry is that of two trees glued to one another with a space-filling curve snaking in between them. In this talk, we first recall a few examples of this, namely, Brownian geometry, Liouville quantum gravity, and the Brownian web. Subsequently, we discuss the construction of a pair of interlaced trees and the corresponding Peano curve in the directed landscape, the conjectural universal scaling limit of models in the Kardar-Parisi-Zhang universality class. Finally, we look at the question of determining the precise Holder and variation regularity of this space-filling curve and discuss some of the ideas involved in the proof. Based on the works arxiv:2304.03269 (joint with Riddhipratim Basu) and arxiv:2301.07704.
2024. 4. 25. (Thu) 17:00-18:00 Room Online
Shledon Dantas (University of Valencia)

Norm-attaining lattice homomorphisms

ABSTRACT. In this talk, we will be discussing the structure of the set ${\rm Hom}(X, \mathbb{R})$ of all lattice homomorphisms from a Banach space $X$ into $\mathbb{R}$ in terms of norm-attaining elements. It turns out that every lattice homomorphism defined on classical Banach lattices ($c_0$, $L^p$-spaces, and $C(K)$-spaces) attains its norm, which shows, in particular, that there is no James theorem for this class of functions. We also discuss the existence of a lattice homomorphism which does not attain its norm and also Bishop-Phelps type theorems in the Banach lattice setting.
2024. 4. 25. (Thu) 16:00-17:00 Room Online
Linjie Zhao (Huazhong University of Science and Technology)

Equilibrium perturbations for stochastic interacting systems

ABSTRACT. In this talk, we consider equilibrium perturbations for two specific interacting particle systems: the generalized exclusion processes and the one-dimensional chain of anharmonic oscillators. By adding a small perturbation to the equilibrium profile and speeding up the process under proper scaling, we show the perturbed quantities evolve according to the Burgers equation in the exclusion process, and to two decoupled Burgers equations in the anharmonic chain, both in the smooth regime. The proof uses the relative entropy method. The talk is based on joint work with Lu Xu.
2024. 4. 23. (Tue) 17:00-18:00 Room Online
Melissa Tacy (University of Auckland)

A quasimode approach to spectral multipliers

ABSTRACT. A central question of Euclidean harmonic analysis is; when does a multiplier $$ Mf = \mathcal{F}^{-1}[m(\cdot)\mathcal{F}[f](\cdot)] $$ defined as an operator $L^2 \to L^2$ extend to a bounded operator $L^p \to L^q$? The Bochner-Riesz multipliers where $$ m_{R}(\xi)=\left(1-\frac{|\xi|^{2}}{R}\right)_{+}^{\delta} $$ are one well-known example of these type of operators. On manifolds we can consider analogous questions about whether spectral multipliers $M(\sqrt{\Delta})$ are bounded. Such questions have been long known to be connected to the growth properties of quasimodes (approximate solutions) to the eigenfunction equation $\sqrt{\Delta}u=\lambda u$. In this talk we will see how we can formalise the relationship between growth properties of quasimodes and boundedness of spectral multipliers and use the relationship to obtain new results about the latter.
2024. 4. 5. (Fri) 13:30-15:00 Room 1423
Kyeongsik Nam (KAIST)

Critical level set percolation of Gaussian Free Field

ABSTRACT. Alexander and Orbach (AO) in 1982 conjectured that the simple random walk on critical percolation clusters exhibit mean field behavior; for instance, its spectral dimension is $4/3$ regardless of the underlying dimension. In a breakthrough work, Kozma and Nachmias established the AO conjecture for critical bond percolation for the (spread out) lattice with dimension $d>6$. In this talk, I will talk about the AO conjecture for the critical level set of Gaussian Free Field, a canonical dependent percolation model of central importance.
2024. 4. 4. (Thu) 16:00-17:00 Room 1423
Jaehwan Choi (KAIST)

Sobolev regularity theory for stochastic reaction-diffusion equations with spatially homogeneous colored noise

ABSTRACT. In this talk, I will discuss results on the existence, uniqueness and Sobolev regularity of solutions to stochastic reaction-diffusion equations (SRDEs) with spatially homogeneous colored noise. I'll start by explaining the concepts of random noise and how solutions to SRDEs are defined. This talk is based on joint works with Beom-Seok Han, Daehan Park, and Jinsol Seo.
2024. 3. 29. (Fri) 16:00-17:30 Room 1423
Seong-Mi Seo (Chungnam National University)

Free energy expansions of 2D Coulomb gas ensembles

ABSTRACT. In this talk, I will discuss recent results on the free energy of logarithmically interacting charges in the plane in an external field. Specifically, at a particular inverse temperature $\beta=2$, this system exhibits the distribution of eigenvalues of certain random matrices, forming a determinantal point process. I will explain how the large $N$ expansion of the free energy depends on the geometric and topological properties of the region where particles condensate, considering the disk, annulus, and sphere cases. I will further discuss the conditional Ginibre ensemble as a non-radial example confirming the Zabrodin-Wiegmann conjecture regarding the spectral determinant emerging at the $O(1)$ term in the free energy expansion. This talk is based on joint works with Sung-Soo Byun, Meng Yang, and Nam-Gyu Kang.
2024. 3. 15. (Fri) 16:00-17:10 Room 1423
Sanghyuk Lee (Seoul National University)

Endpoint bounds on the Hermite spectral projection

ABSTRACT. This talk concerns $L^2$-$L^q$ bounds on the Hermite spectral projection operator $\Pi_\lambda$ in $\mathbb{R}^d$. For $d\ge2$, the optimal $L^2$-$L^q$ bounds on $\Pi_\lambda$ have been known except for the endpoint case $q=2(d+3)/(d+1)$. However, the endpoint bound has been left unsettled for a long time. We prove this missing endpoint case for every $d\ge3$. Our result is based on a new phenomenon: improvement of bounds due to asymmetric localization near the sphere.
2024. 3. 8. (Fri) 16:00-17:30 Room 1423
Sungsoo Byun (Seoul National University)

Free energy expansions of a conditional GINUE and large deviations of the smallest eigenvalue of the LUE

ABSTRACT. In this talk, I will present the free energy expansion of a conditional Ginibre ensemble, which provides the first non-radially symmetric example confirming the Zabrodin-Wiegmann conjecture. After introducing our results from the viewpoint of moments of the characteristic polynomial, I will discuss our findings in the context of the large deviation probabilities of the smallest eigenvalue of the Laguerre unitary ensemble. This talk is based on joint work with Seong-Mi Seo and Meng Yang.
2024. 3. 7. (Thu) 10:00-11:00 Room 1423
Jung-Kyoung Lee (KIAS)

Mixing property of Langevin dynamics

ABSTRACT. The behavior of stochastic systems often varies based on the number of equilibria. A system with a unique equilibrium is known to exhibit a cut-off phenomenon, while a system with multiple equilibria demonstrates metastability. In the context of the mixing property, the former implies fast mixing, whereas the latter implies slow mixing. Until now, considerable research has been dedicated to studying the mixing properties of Langevin dynamics with a unique equilibrium, which exhibits the cut-off phenomenon. The recently developed theory of metastability has enabled us to understand metastable behavior in the context of mixing. In this talk, we study the mixing property of Langevin dynamics with multiple equilibria.
2024. 2. 29. (Thu) 17:30-18:30 Room Online
Olesia Zavarzina (V. N. Karazin Kharkiv National University)

Around the plasticity problem

ABSTRACT. A metric space $M$ is called Expand-Contract plastic (or just plastic) if every non-expansive bijection $F:M\to M$ is an isometry. An intriguing open question is whether the closed unit ball of every Banach space is plastic as a metric space with the metric generated by the norm. There are only several particular results in this direction. The talk will be concentrated on these results, and also on some natural generalizations of plactisity problem and related questions.
2024. 2. 28. (Wed) 15:00-16:00 Room 1423
Marielle Simon (Université Lyon 1)

Energy transport in harmonic chains of oscillators

ABSTRACT. It has been recently observed that some physical or biological systems which are maintained in a bath of constant temperature can behave in an unexpected way: in some cases the temperature stationary profile presents a maximum inside the system higher than the thermostats temperatures, as well as the possibility of uphill diffusion (energy current against the temperature gradient). This is the case for instance in mitochondria, which are present in nearly all types of human cell. In a collaborative work with T. Komorowski and S. Olla, we derive rigourously this "heating inside the system" phenomenon from a microscopic infinite chain of coupled oscillators in contact at both ends with heat baths at different temperatures, and subject to an external force at one end. While heat flows from the thermostats, the mechanical energy produced by the force is then transformed into thermal energy by the bulk dynamics. We follow an approach based on Wigner distributions, which permit to control the energy distribution over various frequency modes and provide a natural separation between mechanical and thermal energies.
2024. 2. 23. (Fri) 15:00-16:00 Room 1424
Marielle Simon (Université Lyon 1)

A few scaling limits results for the facilitated exclusion process, Lecture III

ABSTRACT. I will present some recent results which have been obtained for the facilitated exclusion process, in one dimension. This stochastic lattice gas is subject to strong kinetic constraints which create a continuous phase transition to an absorbing state at a critical value of the particle density. If the microscopic dynamics is symmetric, its macroscopic behavior, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to free boundary problems (or Stefan problems). One of the ingredients is to show that the system typically reaches an ergodic component in subdiffusive time. When the particle system is put in contact with reservoirs of particles (which can either destroy or inject particles at both boundaries), we observe an usual impact on the boundary values of the empirical density. The asymmetric case can also be fully treated (for the infinite particle system): in this case, the empirical density converges to the unique entropy solution to a hyperbolic Stefan problem. All these results rely, to various extent, on a mapping argument with a zero-range process, which completely fails in dimension higher than 1. I will finally discuss some open problems and questions, especially in dimension 2. Based on joint works with O. Blondel, H. Da Cunha, C. Erignoux, M. Sasada and L. Zhao.
2024. 2. 23. (Fri) 13:30-14:30 Room 1424
Marielle Simon (Université Lyon 1)

A few scaling limits results for the facilitated exclusion process, Lecture II

ABSTRACT. I will present some recent results which have been obtained for the facilitated exclusion process, in one dimension. This stochastic lattice gas is subject to strong kinetic constraints which create a continuous phase transition to an absorbing state at a critical value of the particle density. If the microscopic dynamics is symmetric, its macroscopic behavior, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to free boundary problems (or Stefan problems). One of the ingredients is to show that the system typically reaches an ergodic component in subdiffusive time. When the particle system is put in contact with reservoirs of particles (which can either destroy or inject particles at both boundaries), we observe an usual impact on the boundary values of the empirical density. The asymmetric case can also be fully treated (for the infinite particle system): in this case, the empirical density converges to the unique entropy solution to a hyperbolic Stefan problem. All these results rely, to various extent, on a mapping argument with a zero-range process, which completely fails in dimension higher than 1. I will finally discuss some open problems and questions, especially in dimension 2. Based on joint works with O. Blondel, H. Da Cunha, C. Erignoux, M. Sasada and L. Zhao.
2024. 2. 22. (Thu) 15:00-16:00 Room 1423
Marielle Simon (Université Lyon 1)

A few scaling limits results for the facilitated exclusion process, Lecture I

ABSTRACT. I will present some recent results which have been obtained for the facilitated exclusion process, in one dimension. This stochastic lattice gas is subject to strong kinetic constraints which create a continuous phase transition to an absorbing state at a critical value of the particle density. If the microscopic dynamics is symmetric, its macroscopic behavior, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to free boundary problems (or Stefan problems). One of the ingredients is to show that the system typically reaches an ergodic component in subdiffusive time. When the particle system is put in contact with reservoirs of particles (which can either destroy or inject particles at both boundaries), we observe an usual impact on the boundary values of the empirical density. The asymmetric case can also be fully treated (for the infinite particle system): in this case, the empirical density converges to the unique entropy solution to a hyperbolic Stefan problem. All these results rely, to various extent, on a mapping argument with a zero-range process, which completely fails in dimension higher than 1. I will finally discuss some open problems and questions, especially in dimension 2. Based on joint works with O. Blondel, H. Da Cunha, C. Erignoux, M. Sasada and L. Zhao.
2024. 2. 15. (Thu) 17:00-18:00 Room Online
Triinu Veeorg (University of Tartu)

A relative version of Daugavet points and Daugavet property

ABSTRACT. We say that a norm one element $x$ in a Banach space $X$ is a relative Daugavet-point if there exist $\alpha>0$ and $x^*\in S_{X^*}$ with $x^*(x)=1$ such that $\sup_{y\in T}\|x-y\|=2$ for every slice $T$ of $B_X$ which is contained in the slice $S(x^∗,\alpha)$. We show that these points lie strictly between Daugavet points and $\Delta$-points. Furthermore, we provide a condition that a space with the Radon-Nikod\'ym property must satisfy in order to be able to contain relative Daugavet points. We also study these points in absolute sums of Banach spaces and prove that the relative Daugavet property is different from both the Daugavet property and the diametral local diameter 2 property. The talk is based on joint work with T. A. Abrahamsen, R. Aliaga, V. Lima, A. Martiny, Y. Perreau, and A. Prochazka.
2024. 2. 14. (Wed) 16:00-17:10 Room 1423
Jongmin Han (Kyung Hee University)

Nontopological solutions for the self-dual Einstein-Maxwell-Higgs model on a closed surface

ABSTRACT. We consider the self-dual Einstein-Maxwell-Higgs model that is a $U(1)$ gauge field model in the general relative frame. There are three kinds of vortex solutions as the coupling parameter tends to zero: topological solutions, type I & type II nontopological solutions. In this talk, we find a family of type I nontopological solutions by the topological degree method.
2024. 1. 31. (Wed) 17:00-18:00 Room Online
Federico Sau (University of Trieste)

Averaging process on graphs

ABSTRACT. The averaging process is a Markovian mass redistribution model over the vertex set of a connected graph, in which masses sitting on nearest neighbor vertices get deterministically averaged out at random Poissonian times. Despite the latter randomness, this degenerate update rule drives, in the long run, the mass profile towards a deterministic flat configuration. This equilibrium state is also referred to as consensus when interpreting the averaging process as a basic model of opinion dynamics/gossip algorithm, in which vertices represent agents and masses their opinions. The mathematical interest in the model was recently revived by Aldous and Lanoue (2012), later followed by Chatterjee, Diaconis, Sly and Zhang (2022), Quattropani and Sau (2023), and Movassagh, Szegedy and Wang (2022+), all these works being concerned with determining sharp convergence rates to equilibrium for the process. In this talk, we review some basic properties of the averaging process, and present some recent results on mixing times and cutoff for this model. If time permits, we also discuss some recent developments on scaling limits of the process, particularly interesting in this degenerate setting due to the lack of a natural notion of local equilibrium. Based on joint works with Matteo Quattropani (Rome, Sapienza) and Pietro Caputo (RomaTre).
2024. 1. 18. (Thu) 14:00-15:30 Room 1423
Junhee Ryu (Korea University)

Nonlocal elliptic and parabolic equations with general stable operators in weighted Sobolev spaces

ABSTRACT. In this talk, we will discuss nonlocal elliptic and parabolic equations on C^{1,tau} open sets in weighted Sobolev spaces, where tau in (0,1). The operators we consider are infinitesimal generators of symmetric stable L'evy processes, whose L'evy measures are allowed to be very singular. Additionally, for parabolic equations, the measures are assumed to be merely measurable in the time variable. This talk is based on a joint work with Hongjie Dong.
2024. 1. 11. (Thu) 17:00-18:00 Room Online
Andrés Quilis (Université de Franche-Comté)

Retractional local structure of metric spaces

ABSTRACT. In the category of metric spaces, where Lipschitz maps are the natural morphisms, Lipschitz retractions play a central role in the study of the structure of a given metric space. However, in some contexts, the existence of Lipschitz retractions is an overly restrictive property. In this case, it is natural to consider different notions that at least still determine the local structure of metric spaces. In this talk, we take a look at such local concepts: In particular, we define local retracts and almost isometric local retracts. These notions parallel and extend the linear concepts of local complements and almost isometric ideals, considered and studied by authors such as Fakhoury, Kalton (for local complements), and Abrahamsen, Lima, and Nygaard (for almost isometric ideals). We will study the interplay between these two newly introduced concepts and the extension of (isometric) Lipschitz maps, applying the results to the theory of Lipschitz-free spaces. We will characterize as well when a complete metric space is an (almost isometric) local retract in every metric space containing it. Finally, we will improve existing results regarding the existence of a rich almost isometric local retractional structure in every metric space, which heavily contrasts with the situation for the classical notion of Lipschitz retractions.
2024. 1. 11. (Thu) 16:00-17:00 Room 1424
Chamsol Park (Johns Hopkins University)

Logarithmic improved eigenfunction restriction estimates on curves with nonvanishing geodesic curvatures

ABSTRACT. In this talk, we talk about eigenfunctions of the Laplace-Beltrami operator on compact Riemannian manifolds (without boundary). One way to study them is to consider the estimates of the eigenfunctions restricted to submanifolds. We briefly summarize previous results and would like to talk about the logarithmic improved analogue for the restriction to curves with nonvanishing geodesic curvatures, in the presence of nonpositive sectional curvatures in the manifolds. If time permits, we will talk about related future work.
2024. 1. 11. (Thu) 16:00-17:10 Room 1423
Hyunchul Park (SUNY New Paltz)

Spectral heat content for isotropic Lévy processes

ABSTRACT. The spectral heat content (SHC) measures the total heat that remains on a domain when the initial temperature is one and the outside temperature is identically zero. When one replaces the Laplace operator in the heat equation with generators of Lévy processes, one obtains SHC for those Lévy processes. Recently, the two-term asymptotic behavior of SHC for isotropic stable processes on bounded C^{1,1} open sets was investigated by Park and Song (EJP 2022). In this talk, we generalize their result to cover Lévy processes with regularly varying characteristic exponent with index in (1,2]. The proof provides a unified approach to the study of SHC and applies to both Brownian motions and jump processes. This is a joint work with Kei Kobayashi (Fordham University).
2024. 1. 10. (Wed) 10:00-11:30 Room 1423
Jinyeop Lee (Ludwig Maximilian University of Munich)

Second Order Expansion of Gibbs State Reduced Densities in the Gross-Pitaevskii Regime

ABSTRACT. In the seminar, we will explore recent research on the 'Second Order Expansion of Gibbs State Reduced Densities in the Gross-Pitaevskii Regime.' Collaborating with Christian Brennecke and Phan Thành Nam, we investigate quantum fluctuations in a system of $N$ bosons confined to a three-dimensional torus ($\mathbb{T}^3$) with repulsive two-body interactions, emphasizing the large $N$ limit. During the seminar, we will revisit the notions of a Bose gas at zero temperature, including the thermodynamic limit, Gross-Pitaevskii limit, and Lee-Huang-Yang formula. Then we talk about the meaning of positive temperature for a quantum system, including the partition function, Gibbs state. We will then discuss the second-order expressions for one- and two-particle reduced density matrices of the Gibbs state at positive temperatures. This result justifies Bogoliubov's predictions on fluctuations around the condensate.
2024. 1. 4. (Thu) 16:00-17:00 Room 1423
Soobin Cho (University of Illinois Urbana-Champaign)

Heat kernel estimates for fractional Laplacian with supercritical killing potentials

ABSTRACT. This presentation explores heat kernel estimates concerning $−(−\nabla)^{\alpha/2}+V$, where $\alpha\in(0,2)$ and the potential $V$ exhibits singularity at the origin, fitting within the domain of supercritical killing for the fractional Laplacian. Examples include potentials such as $|x|^{−\beta}$ with $\beta>\alpha$. The discussion offers a concise overview of results for both sub-critical and critical scenarios, emphasizing essential distinctions. The primary focus lies in introducing a probabilistic proof, emphasizing the approximate factorization formula for Dirichlet heat kernel estimates. Additionally, time permitting, considerations for large-time estimates will be addressed.
2024. 1. 4. (Thu) 15:00-16:00 Room 1423
Daesung Kim (Georgia Institute of Technology)

Curie-Weiss Model under Lp constraint

ABSTRACT. We consider the Ising Curie-Weiss model on the complete graph constrained under a given $\ell_p$ norm. For $p=\infty$, it reduces to the classical Ising Curie-Weiss model. We prove that for all $p\ge2$, there exists a critical inverse temperature $\beta_c(p)$ such that for $\beta<\beta_c(p)$, the magnetization is concentrated at zero and satisfies an appropriate Gaussian CLT. On the other hand, for $\beta>\beta_c(p)$, the magnetization is not concentrated at zero similar to the classical case. We further generalize the model for general symmetric spin distributions and prove similar phase transition. In this talk, we discuss a brief overview of classical Curie-Weiss model, a generalized Hubbard-Stratonovich transforms, and how we apply the transform to Curie-Weiss model under $\ell^p$ constraint. This is based on joint work with Partha Dey.
2024. 1. 2. (Tue) 16:00-17:10 Room 1423
Yilin Wang (IHES)

Holography of the Loewner energy

ABSTRACT. The Loewner energy is a Möbius invariant quantity that measures the roundness of Jordan curves on the Riemann sphere. It is the Onsager-Machlup action of SLE loop measure and is also a Kähler potential on the Weil-Petersson Teichmüller space. Motivated by AdS/CFT correspondence and the fact that Möbius transformations extend to isometries of the hyperbolic 3-space H^3, we look for a quantity defined geometrically in H^3 which equals the Loewner energy of a Jordan curve in the conformal boundary. We show that the Loewner energy equals the renormalized volume of a submanifold of H^3 constructed using the Epstein surfaces associated to the hyperbolic metric on both sides of the curve. This is a joint work with Martin Bridgeman (Boston College), Ken Bromberg (Utah), and Franco Vargas-Pallete (Yale).

Past Seminars

2023 Seminars
2022 Seminars
2021 Seminars
2020 Seminars
2019 Seminars

Organizers

Yong-Gwan Ji - Korea Institute for Advanced Study
Mingu Jung - Korea Institute for Advanced Study
Myeongju Kang - Korea Institute for Advanced Study
Nam-Gyu Kang - Korea Institute for Advanced Study
Junha Kim - Korea Institute for Advanced Study
Seonwoo Kim - Korea Institute for Advanced Study
Joonhyun La - Korea Institute for Advanced Study
Jaehun Lee - Korea Institute for Advanced Study
Jaehun Lee - KAIST
Jung-Kyoung Lee - Korea Institute for Advanced Study
Taehun Lee - Korea Institute for Advanced Study
Sewook Oh - Korea Institute for Advanced Study
Sung-Jin Oh - UC Berkeley & KIAS
Hyangdong Park - Korea Institute for Advanced Study
Jaehyeon Ryu - Korea Institute for Advanced Study
Jinsol Seo - Korea Institute for Advanced Study
Hyungsung Yun - Korea Institute for Advanced Study

Directions

Korea Institute for Advanced Study
Buildings 1 & 8, 85 Hoegi-ro, Dongdaemun-gu, Seoul, Korea

June E Huh Center for Mathematical Challenges
2nd Floor, 118 Hongneung-ro, Dongdaemun-gu, Seoul, Korea