Analysis, PDE & Probability Seminar

2024 Seminars

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

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

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