Analysis, PDE & Probability Seminar

Analysis, PDE & Probability Seminar

2025. 3. 28. (Fri) 17:00-18:00 Room Online
Abraham Rueda Zoca (University of Granada)

On some topological properties in tensor products of Banach spaces

ABSTRACT. We will expose the part of the work [1] where we study several topological properties (like weakly compactly generated, weakly Lindelöf determined, property (C) of Corson...) on tensor products of Banach spaces. [1] A. Avil\'es, G. Mart\'inez-Cervantes, J. Rodr\'iguez and A. Rueda Zoca, Topological properties in tensor products of Banach spaces, J. Funct. Anal. 283 (2022), article 109688.
2025. 3. 28. (Fri) 11:00-12:00 Room 1423
Minkyu Lim (KAIST SAARC)

Gradient estimates of very weak solutions to mixed local problems

ABSTRACT. The mixed problem, which involves an elliptic equation with both local and nonlocal terms, has recently gained attention. It arises from the combination of two stochastic processes with different scales, such as a classical random walk and a Lévy process. In this presentation, we provide a sub-natural gradient estimate for a very weak solution to the mixed problem, under the condition that the integrability of the source term is below the natural exponent.
2025. 3. 28. (Fri) 11:00-12:00 Room 204 (Soorim)
Seonwoo Kim (Yonsei University)

$\Gamma$-Expansion of the Measure-Current Large Deviations Rate Functional

ABSTRACT. In this talk, we present the general theory of connecting the hierarchical structure of metastability and the $\Gamma$-expansion of the measure-current (or level-2.5) large deviations rate functional of non-reversible finite-state Markov chains. The first bridge between the metastability and the $\Gamma$-expansion of the (measure, or level-2) rate functional was discovered recently by Bertini, Gabrielli and Landim (Ann. Appl. Probab. 2024). We briefly review the previous works on this topic and present our own contributions. The talk is based on a recent work (arXiv:2412.13515) with Claudio Landim (IMPA).
2025. 3. 24. (Mon) 14:00-15:00 Room Online
Jongmyeong Kim (Academia Sinica)

Existence of one point singular symmetric self-similar solution of fast diffusion equation and Its large time behavior

ABSTRACT. In this talk, I will present the existence of global, forward eternal, one point singular, symmetric, and self-similar solution of fast diffustion equation u_t = \Delta u^m for n>2 and 0
2025. 3. 21. (Fri) 16:00-17:00 Room 1423
Eui Yoo (Seoul National University)

Three topological phases of the elliptic Ginibre ensembles with a point charge

ABSTRACT. In the large N limit, complex and symplectic random matrix models exhibit limiting spectra whose support is called the droplets. In this talk, we will discuss the elliptic Ginibre ensemble conditioned to have real eigenvalue with multiplicity proportional to the dimension of the matrix. We prove that the droplets are either simply connected, doubly connected, or composed of two simply connected components. Moreover, we present the explicit description of the droplet and electrostatic energies for the simply and doubly connected case. Finally, we introduce the asymptotic behavior of the moments of the characteristic polynomials of elliptic Ginibre matrices as an application. This is a joint work with Sung-Soo Byun.
2025. 3. 12. (Wed) 11:00-12:00 Room 1423
Minhyun Kim (Hanyang University)

Removable singularities for nonlocal equations and nonlocal minimal graphs

ABSTRACT. TBA
2025. 2. 27. (Thu) 16:00-17:00 Room 1503
Jeongtae Oh (Seoul National University)

Multiparameter Lacunary Maximal Functions on the Heisenberg Group

ABSTRACT. Ricci and Stein, in their work, considered the multiparameter lacunary maximal operator under suitable regularity conditions on the measure and established L^p estimates. In this talk, we introduce the Lacunary Elliptic Maximal Operator on the Heisenberg group and outline how to prove its L^p boundedness. Finally, under the regularity condition proposed by Ricci and Stein, we discuss how these results can be extended in the Heisenberg group setting.
2025. 2. 20. (Thu) 11:00-12:00 Room 1423
Takwon Kim (Sungshin Women's University)

Higher Regularity of Degenerate or Singular Equations via Generalized Schauder Theory

ABSTRACT. In this presentation, we introduce generalized Schauder theory for degenerate or singular parabolic equations and propose a novel approach to overcoming the limitations of traditional regularity theory. Conventional regularity methods often rely on the bootstrap argument to establish higher-order regularity, but this approach is not applicable to degenerate or singular equations. To address this challenge, we develop a new approximation method based on s-polynomials, which approximate the coefficients of the equation using fractional-order monomials rather than classical polynomials. Using this framework, we establish C_s^{2,α}-regularity for solutions and provide a theoretical foundation for proving higher regularity even in the presence of degeneracy or singularity. Furthermore, we demonstrate that this generalized Schauder theory is applicable to various problems, including Monge–Ampère equations, nonlocal equations, Gauss curvature flow, and Porous medium equations.
2025. 2. 14. (Fri) 16:30-17:30 Room Online
María del Mar González (Autonomous University of Madrid)

The limiting case of the fractional Caffarelli-Kohn-Nirenberg inequality in dimension one

ABSTRACT. We study the limiting case in dimension one for the fractional Caffarelli-Kohn-Nirenberg inequality, obtaining Onofri’s inequality in the unit disk as a limit. One difficulty here is the lack of an explicit expression for the extremal. An important aspect is the study of solutions of the weighted Liouville equation for the half-Laplacian in dimension one. This is joint work with A. Hyder (TIFR Bangalore) and M. Saez (U. Pontificia Chile).
2025. 2. 6. (Thu) 10:30-11:30 Room 1423
Ho-Sik Lee (Universität Bielefeld)

Regularity results for nonlocal equations

ABSTRACT. By Green's representation formula, the gradient of the solution to Poisson's equation is pointwisely bounded by a Riesz potential of the right-hand side of the equation, the so-called gradient potential estimate. We show gradient potential estimates for nonlinear nonlocal elliptic equations from the joint with Prof. Lars Diening (Bielefeld), Kyeongbae Kim (SNU), and Dr. Simon Nowak (Bielefeld). Next, we consider examples of solutions that such higher differentiability estimates fail when the nonlocal equation has irregular coefficients, even in the homogeneous linear case. Inspired by Meyers' example, which is considered in [Meyers '63] and [Kh. Balci, Diening, Giova, Passarelli di Napoli '22], we give such examples of nonlocal equations. This is the joint work with Prof. Anna Kh. Balci (Charles), Prof. Lars Diening, and Prof. Moritz Kassmann (Bielefeld).
2025. 1. 24. (Fri) 17:00-18:00 Room Online
Colin Petitjean (Gustave Eiffel University)

On curve-flat Lipschitz functions and their linearizations

ABSTRACT. Given two pointed metric spaces $M$, $N$, any basepoint-preserving Lipschitz map $f:M \to N$ can be associated with a canonical linearization $T_f:\mathcal{F}(M) \to \mathcal{F}(N)$, which is a bounded linear operator. The spaces $\mathcal{F}(M)$ and $\mathcal{F}(N)$ are known as Lipschitz free spaces over $M$ and $N$, respectively. A more or less recent trend explores the relationship between the properties of the nonlinear map $f$ and those of the linear operator $T_f$. In this talk, we will characterize the Lipschitz maps $f$ for which $T_f$ is a Dunford-Pettis operator (also referred to as completely continuous). If time permits, we will also discuss additional operator properties equivalent to the Dunford-Pettis property within this specific framework. This work is based on a collaboration with Gonzalo Flores, Mingu Jung, Gilles Lancien, Antonin Prochazka, and Andrés Quilis.
2025. 1. 23. (Thu) 15:00-16:00 Room 204 (Soorim)
Joonil Kim (Yonsei University)

Oscillatory integrals over global domains

ABSTRACT. Since Varchenko's seminal work, the asymptotics of oscillatory integrals and related problems have been effectively analyzed using the Newton polyhedra associated with the phase function $P$. These analyses have traditionally focused on integrals supported in sufficiently small neighborhoods. The purpose of this talk is to introduce the estimates of sub-level sets and oscillatory integrals over global domain $D$ with a primary focus on $D=\mathbb{R}^d$. In particular, we shall discuss (i) Piuseux series on $\mathbb{R}^2$ and resolutions of singularities, (ii) Analogue of Varchenko's theorem on the global domain (iii) Strichartz type inequality for dispersive partial differential equations (PDEs) as a key application. (iv) Phong and Stein's theorem regarding the global oscillatory integral operator (a joint work with Minbeom Kang)

Past Seminars

2024 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
Nam-Gyu Kang - Korea Institute for Advanced Study
Seonwoo Kim - Korea Institute for Advanced Study
Joonhyun La - Korea Institute for Advanced Study
Jung-Kyoung Lee - Korea Institute for Advanced Study
Juyoung Lee - Korea Institute for Advanced Study
Se-Chan 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