Analysis, PDE & Probability Seminar

Upcoming Seminars

2026. 4. 6 (Mon) 16:00-17:00  Room 1424
Youngjoon Hong (Seoul National University)

Approximation and generalization in multilayer perceptrons and their applications

ABSTRACT. This talk presents a mathematical overview of machine learning and deep learning, with a focus on approximation and generalization for multilayer perceptrons. We begin with a high-level map of the main mathematical ingredients behind modern learning theory. We then discuss approximation results for MLPs from a function-space perspective and highlight what these results do and do not explain about practical performance. Next, we introduce several representative frameworks for understanding generalization, including Rademacher complexity, and discuss how gradient-based training interact with these theories. If time permits, we briefly survey how mathematical ideas appear in modern generative modeling—such as diffusion models—through connections to stochastic differential equations and their numerical schemes. The goal is to clarify how theoretical principles relate to practical research in deep learning and to identify open mathematical challenges.

2026. 4. 7 (Tue) 16:00-17:00  Room 1423
Uihyeon Jeong (KAIST)

Quantized slow blow-up dynamics for the energy-critical corotational wave map problem

ABSTRACT. In this talk, we consider the blow-up dynamics of co-rotational solutions for energy-critical wave maps with the 2-sphere target. We briefly introduce the (2+1)-dimensional wave maps problem and its co-rotational symmetry, which reduces the full wave map to the (1+1)-dimensional semilinear wave equation. Under such symmetry, we see that this problem has a unique explicit stationary solution, so-called "harmonic map". Then we point out some of the works of analyzing the long-term dynamics of the flow near the harmonic map. Among them, we focus on the smooth blow-up result that corresponds to the stable regime. In particular, the case where the homotopy index is one has a distinctive nature from the other cases, which allows us to exhibit the smooth blow-up with the quantized blow-up rates corresponding to the excited regime.

2026. 4. 10 (Fri) 16:00-17:00  Room 1424
Jinwoo Jang (POSTECH)

Long-Time Dynamics of the 3D Vlasov–Maxwell System with Boundaries

ABSTRACT. In this seminar, we study the Vlasov–Maxwell system, a fundamental collisionless kinetic model for plasmas, posed in a three-dimensional half-space with boundaries. We begin with a brief warm-up by revisiting the one-dimensional Vlasov–Poisson system in the absence of magnetic fields, focusing on Penrose’s classical 1960 spectral criterion for linear stability and instability. We then turn to the full Vlasov–Maxwell system and discuss the major analytical difficulties introduced by electromagnetic coupling, boundary effects, and nonlinear interactions. In particular, we highlight the role of an effective gravitational force directed toward the boundary and its interplay with boundary temperature conditions. This viewpoint naturally leads us to formulate a conjectural linear instability criterion associated with boundary-induced confinement effects. Within this framework, we construct global-in-time classical solutions to the nonlinear Vlasov–Maxwell system beyond the vacuum scattering regime. Our approach combines the construction of stationary boundary equilibria with a proof of their asymptotic stability in the $L^\infty$ setting under small perturbations. This work provides a new framework for analyzing long-time plasma dynamics in bounded domains with interacting magnetic fields. To our knowledge, it yields the first construction of asymptotically stable non-vacuum steady states for the full three-dimensional nonlinear Vlasov–Maxwell system. This is joint work with Chanwoo Kim.

2026. 4. 17 (Fri) 15:00-16:00  Room 1423
Seunghwan Hwang (Seoul National University)

Construction of infinite time bubble tower solutions to critical wave maps equation

ABSTRACT. This is the first part of a joint presentation with Kihyun Kim (Seoul National University), on our recent joint work on the construction of infinite time bubble tower solutions to the critical wave maps equation taking values in the two-sphere. In the first part, I will present the background, the main result, and a brief sketch of the proof. Our main result constructs, for any integers k\geq3 and J\geq1, a solution that is global in one time direction, has k-corotational symmetry, and asymptotically decomposes into J-many concentric bubbles of alternating signs with asymptotically vanishing radiation. The scales of each bubble are of order t^{-\alpha_{j}} with \alpha_{j}=(\frac{k}{k-2})^{j-1}-1. This shows the existence of multi-bubble solutions with an arbitrary number of bubbles in soliton resolution, provided that k\geq3, global existence in one time direction, and alternating signs are considered. Our proof is based on modulation analysis with the method of backward construction. The key new ingredient is a Morawetz-type functional that provides suitable monotonicity estimates for solutions around multi-bubble configurations.

2026. 4. 17 (Fri) 16:00-17:00  Room 1423
Kihyun Kim (Seoul National University)

Construction of infinite time bubble tower solutions to critical wave maps equation

ABSTRACT. This is the second part of a joint presentation with Seunghwan Hwang (Seoul National University), on our recent joint work on the construction of infinite time bubble tower solutions to the critical wave maps equation taking values in the two-sphere. Having discussed the background, the result, and the scheme of the proof in the previous talk, I will present more details of the proof in this second part of the talk.

2026. 5. 15 (Fri) 16:00-17:30  Room 1424
Junsik Bae (Kyungpook National University)

Traveling waves in the reaction-diffusion equation with discontinuity

ABSTRACT. We classify traveling waves and stationary solutions of a reaction–diffusion equation arising in population dynamics with Allee-type effects. The reaction term is given by a quadratic polynomial with a discontinuity at zero, which captures finite-time extinction for sub-threshold populations. This discontinuity induces a free boundary in the wave profile, a phenomenon that distinguishes the model from the classical logistic or Allen–Cahn equations. A complete scenario is presented that connects monostable and bistable traveling waves through the wave speed parameter. This is a joint work with Wonhyung Choi (Hankyong National Univ.) and Yong-Jung Kim (KAIST).

Past Seminars

2026 Seminars
2025 Seminars
2024 Seminars
2023 Seminars
2022 Seminars
2021 Seminars
2020 Seminars
2019 Seminars

Organizers

Jae-Hwan Choi   -   Korea Institute for Advanced Study

Namhyun Eun   -   Korea Institute for Advanced Study

In-Jee Jeong   -   Korea Institute for Advanced Study

Yong-Gwan Ji   -   Korea Institute for Advanced Study

Nam-Gyu Kang   -   Korea Institute for Advanced Study

Taegyu Kim   -   Korea Institute for Advanced Study

Wontae Kim   -   Korea Institute for Advanced Study

Joonhyun La   -   Korea Institute for Advanced Study

Juyoung Lee   -   Korea Institute for Advanced Study

Se-Chan Lee   -   Korea Institute for Advanced Study

Sewook Oh   -   Korea Institute for Advanced Study

Sung-Jin Oh   -   UC Berkeley & KIAS

Junhee Ryu   -   Korea Institute for Advanced Study

Jinsol Seo   -   Korea Institute for Advanced Study

Young-Jin Sim   -   Korea Institute for Advanced Study

Kyeong Song   -   Korea Institute for Advanced Study

Jaeyun Yi   -   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