Analysis, PDE & Probability Seminar

2026 Seminars

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).

2026. 5. 11 (Mon) 16:00-17:00  Room 1424
Jaeyong Shin (Yonsei University)

Eventual regularity and asymptotic behavior of Leray-Hopf weak solutions for the Hall MHD equations

ABSTRACT. In this talk, we study Leray-Hopf weak solutions for the incompressible, viscous and resistive Hall magnetohydrodynamic (Hall MHD) equations based on a joint work with Jinwook Jung (Hanyang University). We begin with a brief review of the fundamental properties of Leray-Hopf weak solutions for the incompressible Navier-Stokes equations. We then consider the Hall MHD equations depending only on two spatial variables and show that any Leray-Hopf weak solution can become eventually smooth. Based on this eventual smoothness, we derive the asymptotic behavior of Leray-Hopf weak solutions. For the three dimensional case, we construct weak solutions which eventually obtain additional regularities by investigating the magneto-vorticity field structure.

2026. 5. 6 (Wed) 16:00-17:00  Room 1423
Jorge Olivares (Fudan University)

Non-existence of wandering intervals for asymmetric unimodal maps

ABSTRACT. A natural problem in dynamical systems is the classification of “equivalent” systems. In the real one-dimensional setting, the ordering of the phase space (the circle or a compact interval) gives rise to the question of whether the order of the orbits determines the system. More precisely, one may ask whether two maps that are combinatorially equivalent are topologically conjugate. This problem goes back to Poincaré’s work on circle homeomorphisms, and a positive answer depends on the non-existence of wandering intervals. In this talk, I will discuss joint work with Professor Weixiao Shen at Fudan University, where we prove that an asymmetric unimodal map (that is, one with different left and right critical orders) has no wandering intervals.

2026. 4. 30 (Thu) 15:00-16:00  Room 8101
Hyangdong Park (KAIST)

Shocks and contact discontinuities for the steady Euler system

ABSTRACT. There are two types of discontinuous transition phenomena in inviscid compressible flows, shocks and contact discontinuities. We will discuss the existence and stability of shocks and contact discontinuities for the steady Euler system. The Helmholtz decomposition method and the iteration method for 2-D and 3-D axisymmetric flows with nonzero vorticity will be introduced.

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. 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. 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. 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. 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. 1 (Wed) 16:00-17:00  Room 1423
Kyeongsu Choi (KIAS)

Zoo of ancient curve shortening flows

ABSTRACT. We will talk about classification results for the ancient solutions to the curve shortening flows. Also, we introduce some interesting examples of ancient flows as well.

2026. 4. 1 (Wed) 16:00-17:00  Room 1423
In-Jee Jeong (KIAS)

Dynamics of vortex rings

ABSTRACT. We study the dynamics of a single vortex ring of small cross-section in the three-dimensional incompressible Euler equations. For a broad class of initial vorticities concentrated near a vortex ring, we prove that the solution remains sharply localized around a moving core for all times and propagates along its axis with the speed predicted by the Kelvin-Hicks formula. Moreover, we show that such vortex rings are dynamically unstable under arbitrarily small perturbations: suitable smooth perturbations lead to linear-in-time filamentation in the axial direction. These results provide a quantitative description of the coexistence of long-time coherence and instability mechanisms for vortex rings in inviscid flows.

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. 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. 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. 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) 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. 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. 10 (Tue) 11:00-12:00  Room 1423
Jaeyun Yi (KIAS HCMC)

Advection-Diffusion equation driven by random velocity

ABSTRACT. We discuss the long-time behavior of advection-diffusion equations driven by random velocity. It is well-known that the $L_2$ norm of the solution decays at least exponentially fast when the velocity field is incompressible. We know that this exponential decay is optimal in the case of pure diffusion, but is less clear once the advection is added. Establishing lower bounds on the decay rate, especially faster than exponential, has attracted significant attention in both the deterministic and stochastic PDE communities. We show that when the advection-diffusion equation is driven by a “typical random” velocity, the exponential decay is optimal. Our result partially answers the Bachelor conjecture concerning the natural length scale of passive scalars. This is joint work with Martin Hairer, Tommaso Rosati, and Sam-Punshon Smith.

2026. 3. 10 (Tue) 10:00-11:00  Room 1423
Young-Jin Sim (KIAS)

Existence and stability of Sadovskii vortex patch: an odd-symmetric touching pair of uniform vortices

ABSTRACT. As a traveling solution of the two-dimensional incompressible Euler equations, the Sadovskii vortex patch takes the form of a counter-rotating pair of vortex patches that are in contact. This model was suggested by Sadovskii [J. Appl. Math. Mech., 1971] and has since attracted considerable attention due to its significance to the inviscid limit of planar flows via the Prandtl-Batchelor theory and its role as an asymptotic state of the dynamics of vortex rings. In this paper, we establish the existence and stability of the Sadovskii vortex patch. First, we show that a Sadovskii vortex patch arises as a maximizer of the kinetic energy under an exact impulse condition. By analyzing the fluid velocity along the symmetry axis and its relation to the vorticity of a dipole, we verify that two patches in the dipole are in contact. Second, we construct a subcollection of such energy maximizers that is stable in the following sense: if an initial vorticity is sufficiently close to a maximizer, then the corresponding solution remains close to this collection up to a translation and travels at a similar speed to the maximizers. This is achieved via a concentration-compactness argument together with the conservation of impulse, circulation, and kinetic energy in time. Furthermore, using uniform estimates of energy maximizers and a shift estimate obtained by estimating the center of mass of the solution, we show that the solution keeps its touching structure uniformly in time.

2026. 3. 4 (Wed) 16:00-18:00  Room 1424
Ken Abe (Osaka Metropolitan University)

MHS equilibria in the non-resistive limit to the randomly forced resistive magnetic relaxation equations

ABSTRACT. I will discuss the question of the existence of 3D steady Euler flows (MHS equilibria) via magnetic relaxation equations, and introduce our recent study on their statistical model. This talk is based on joint work with I.-J. Jeong (KIAS), F. Pasqualotto (UCSD), and N. Sato (NIFS).

2026. 2. 24 (Tue) 16:00-17:00  Room 8309
Kenta Nakamura (Nihon University)

Volume concentration phenomenon for the p-Sobolev flow

ABSTRACT. We provide a new quantitative local boundedness for the p-Sobolev flow, which is a nonlinear generalization of the gradient flow concerning the p-Dirichlet energy on a bounded domain. In the case p=2, the p-Sobolev flow is the heat flow. This estimate enables us to observe a volume concentration phenomenon for the p-Sobolev flow. This is based on a joint work with Tuomo Kuusi and Masashi Misawa.

2026. 2. 23 (Mon) 16:00-17:00  Room 1423
Hyungsung Yun (Changwon National University)

Boundary Hölder gradient estimates for fully nonlinear degenerate or singular parabolic equations

ABSTRACT. In this talk, we study the boundary regularity of viscosity solutions to fully nonlinear degenerate or singular parabolic equations of the form $$u_t = |Du|^\gamma F(D^2u)+f, \quad \gamma>-1.$$ The gradient-dependent degeneracy or singularity, together with the time derivative, poses significant difficulties compared to the elliptic case. We establish boundary $C^{1,\alpha}$ estimates for the Dirichlet problem under suitable structural assumptions on the operator and boundary data. The proof combines compactness arguments, barrier constructions, regularization techniques, and the boundary regularity of small perturbation solutions, together with the analysis of a model problem on a flat boundary. Our results unify and extend previous boundary regularity results and also provide a framework for studying boundary behavior in a broader class of nonlinear parabolic equations.

2026. 2. 9 (Mon) 14:00-15:30  Room 1423
Jinyeop Lee (University of Basel)

The Semiclassical Limit of the 2D Dirac–Hartree Equation with Periodic Potentials

ABSTRACT. We study the semiclassical limit of the two-dimensional Dirac–Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors onto the positive and negative energy bands. Under suitable assumptions on the initial data and the potentials, we rigorously derive Vlasov-type transport equations describing the evolution of the band-resolved phase-space densities in both the massive and massless regimes. In the massless case, the limiting dynamics propagate ballistically with constant speed, while in the massive case the velocity is relativistic. Our analysis justifies the emergence of relativistic Vlasov equations from Dirac-Hartree dynamics in the semiclassical regime. As a corollary, we recover the relativistic Vlasov-Poisson equation from the Dirac equation with a regularized Coulomb interaction when the regularization vanishes together with the semiclassical parameter. This is joint work with Kunlun Qi.

2026. 2. 3 (Tue) 10:00-11:00  Room Online
Hyunwoo Kwon (Brown University)

표면장력효과를 고려한 단상 무스캣 문제의 대역해의 존재성

ABSTRACT. 단상 무스캣 문제는 다공성 매질에서 건조한 영역과 Darcy's flow에 의해 기술되는 유체가 있는 영역 사이의 경계면의 발전을 다룹니다. 표면장력효과를 고려하는 것은 물리적인 이유로도 자연스럽지만, 이 효과를 고려할 경우, 해 자체에 대해 극대원리와 같은 수단을 쓸 수가 없어 문제해결 난이도가 복잡해집니다. 이 발표에서는 전공간에서 정의되었거나 주기성을 고려해도 초기조건이 적절히 작다면, 대역해의 존재성을 보입니다. 더불어 주기성을 고려할 경우, 해의 점근성도 보입니다. 본 발표는 董弘桀(동홍지에) 교수님과의 공동연구를 바탕으로 하며, 발표자료도 한국어로 진행합니다.

2026. 1. 28 (Wed) 16:00-17:00  Room 1423
Junkee Jeon (Kyunghee University)

Stochastic Control and Free Boundary Problems Arising in Financial Mathematics

ABSTRACT. In this presentation, I discuss stochastic control problems arising from financial mathematics. Specifically, I address problems that combine stochastic control with optimal stopping, as well as those involving singular control and switching control. Furthermore, I investigate a two-person zero-sum game played between a singular controller and an optimal stopper. A key focus of this talk is deriving the associated parabolic variational inequalities and analyzing the analytic properties of the free boundaries they induce. I will also explain how these boundaries characterize the optimal strategies. Finally, I illustrate the application of these models in finance with numerical examples.

2026. 1. 14 (Wed) 16:00-17:00  Room 1423
Kihoon Seong (Cornell University)

Sine–Gordon measures, topological soliton, and Gaussian multiplicative chaos

ABSTRACT. I will discuss the sine–Gordon QFT, which lies in the same universality class as the Coulomb gas and the XY model, and exhibits the BKT phase transition (Nobel prize 2016). The sine–Gordon action admits infinitely many homotopy classes. Even though the action has no energy minimizers in homotopy classes with high topological charge, I will show that the Gibbs measure nevertheless concentrates and exhibits Ornstein–Uhlenbeck fluctuations around the multi-soliton manifold. I will then discuss the geometry of this multi-soliton manifold, including how solitons are arranged, such as their expected locations and gaps. Furthermore, I will explain the asymptotics of the partition function using Gaussian multiplicative chaos. Based on joint work with Hao Shen and Philippe Sosoe.

Past 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

Jung-Kyoung Lee   -   Korea Institute for Advanced Study

Juyoung Lee   -   Korea Institute for Advanced Study

Se-Chan Lee   -   Korea Institute for Advanced Study

Deokwoo Lim   -   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

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

Jonguk Yang   -   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