Analysis, PDE & Probability Seminar

Upcoming Seminars

2026. 3. 16 (Mon) 16:00-17:00  Room 1424
Namhyun Eun (KIAS)

Stability for Oscillatory Shocks of KdV-Burgers Equation

ABSTRACT. It is well known that the Korteweg-de Vries-Burgers (KdVB) equation admits traveling wave solutions, referred to as viscous-dispersive shock waves. When the dispersion dominates viscosity, the associated shock profiles exhibit infinitely many oscillations. Although the KdVB equation serves a a canonical model incorporating nonlinearity, viscosity, and dispersion, the stability of these shock profiles-especially in the oscillatory regime-remains poorly understood. In this talk, we discuss detailed structural properties of the shock profiles, and then prove an L^2 contraction property under arbitrarily large perturbations, up to a time-dependent shift. Moreover, we also justify zero viscosity-dispersion limits. This is a collaborative work with Geng Chen (University of Kansas), Moon-Jin Kang (KAIST), and Yannan Shen (University of Kansas).

2026. 3. 23 (Mon) 14:00-15:00  Room 1424
Geuntaek Seo (POSTECH)

Asymptotic Convergence of Nonconvex Wasserstein Gradient Flows and Interacting Langevin Dynamics

ABSTRACT. Over the past two decades, gradient flows in the optimal transport framework have attracted significant attention due to their broad range of applications. Despite substantial progress in convex settings, a general theory for nonconvex energy functionals remains largely undeveloped. Our aim is to establish the theory of asymptotic convergence of Wasserstein gradient flows for nonconvex functionals on $\mathcal{P}_2(M)$, the set of probability measures on a Riemannian manifold $M$ with finite second moment, by applying Łojasiewicz–Simon theory in its tangent space. In particular, I will present this approach in the McKean–Vlasov setting, including asymptotic convergence results for the equations arising as mean-field limits of interacting Langevin dynamics. This is based on joint work with Beomjun Choi (KAIST) and Seunghoon Jeong (POSTECH).

2026. 3. 23 (Mon) 15:00-16:00  Room 1424
Sungjin Lee (Sogang University)

Two-sided Gaussian estimates for fundamental solutions of second-order parabolic equations in non-divergence form

ABSTRACT. We construct the fundamental solution of second order parabolic operators $P$ in non-divergence form under minimal assumptions, namely that the coefficients are of Dini mean oscillation in the spatial variables. We also establish that the fundamental solution satisfies a two-sided Gaussian estimates. Specifically, we show that the upper and lower bounds follow from the local boundedness property and the weak Harnack inequality for the adjoint operator $P^*$, respectively. This provides a simpler and more direct proof of the Gaussian estimates when the coefficients have Dini mean oscillation in the spatial variables, avoiding the use of normalized adjoint solutions required in previous works. Additionally, we study the regularity of parabolic operators $P$ in non-divergence form and that of its adjoint operator $P^*$.

2026. 3. 23 (Mon) 16:00-17:00  Room 1424
Dong-ha Kim (Ajou University)

Dirichlet problem and regular boundary points for elliptic equations in non-divergence and double divergence form

ABSTRACT. In this talk, we consider the Dirichlet problem for second-order elliptic equations in non-divergence form $L$ and double divergence form $L^*$. We introduce a potential theory framework for these operators, including Perron’s method, capacity theory, and Wiener’s criterion. We establish the equivalence between regular boundary points for these operators $L$ and $L^*$ with those for the Laplace operator, assuming that the leading coefficients satisfy the Dini mean oscillation condition. As a corollary, we also construct the Green’s function for those operators in regular domains.

2026. 3. 26 (Thu) 15:00-16:00  Room 8101
Leah Schätzler (Aalto University)

Nonlinear parabolic partial differential equations in noncylindrical domains

ABSTRACT. Parabolic PDEs are usually posed in a space-time cylinder $\Omega \times [0,T)$ with time $0 < T \leq \infty$ and a fixed spatial domain $\Omega \subset \mathbb{R}^n$. However, both from a theoretical perspective and in view of physical and biological applications, it is interesting to consider sets $\Omega(t)$ that are allowed to change in time. From the viewpoint of space-time, this means that we are dealing with a noncylindrical domain. In this talk, I will present recent existence and global higher integrability results for certain nonlinear parabolic PDEs in such noncylindrical domains. In particular, I will draw connections between the regularity of solutions and the speed at which the spatial domain is allowed to grow or shrink in time. The talk is based on joint work with Kristian Moring, Christoph Scheven, Jarkko Siltakoski, and Calvin Stanko.

2026. 3. 26 (Thu) 16:00-17:00  Room 8101
Theo Elenius (Aalto University)

Carleson-type removability for $p$-parabolic equations

ABSTRACT. We characterize removable sets for Hölder continuous solutions to degenerate parabolic equations of $p$-growth. A sufficient and necessary condition for a set to be removable is given in terms of an intrinsic parabolic Hausdorff measure, which depends on the considered Hölder exponent. We present a new method to prove the sufficient condition, which relies only on fundamental properties of the obstacle problem and supersolutions, and applies to a general class of operators. For the necessity of the condition, we establish the Hölder continuity of solutions with measure data, provided the measure satisfies a suitable decay property. The techniques developed here provide a new point of view even in the case $p=2$. The talk is based on joint work with Michał Borowski, Leah Schätzler, and David Stolnicki.

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)

TBA

ABSTRACT. TBA

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
황승환 (Seoul National University)

TBA

ABSTRACT. TBA

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

TBA

ABSTRACT. TBA

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