Date/Time : 05 May 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Yu-Sen An (Nanjing University of Aeronautics and Astronautics), Yu-Sen An (Nanjing University of Aeronautics and Astronautics)

With the detection of gravitational waves and the release of black hole shadow images, black hole becomes a real astrophysical object instead of a mere mathematical artifact. However, while extensive works have been done on the black hole exterior, the interior region remains far less explored. Thus in this talk, I will focus on the black hole interior structure, from the horizon down to the singularity. We are particularly interested in the black hole spacetime in anti–de Sitter (AdS) space since this can be connected to quantum many body systems through holographic duality. I will show that the interior dynamics are far richer than previously expected. I will choose several important holographic models such as holographic superconductor to illustrate these distinct behaviors. Finally, I will demonstrate how to connect these interior structures to the boundary physics, potentially revealing new aspects of holographic duality.

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Date/Time : 05 May 2026 16:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Nguyen Thac Dung, Hanoi University of Sciences, (VNU)

It is well-known that the Bochner technique is very useful to study geometry and topology of Riemannian manifolds. In this talk, I mention a new theory about curvature conditions related to the Bochner techniques following recent works by Petersen and Wink. I also introduce their applications on compact manifolds. Then, I will extend their results on non-compact complete Riemannian manifolds and prove some vanishing and rigidity results.

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Date/Time : 22 Apr 2026 14:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Jia Shi, Indiana University Bloomington

In this talk, I will introduce the implosion blow-up results for the compressible Euler, the compressible Navier-Stokes equations, and the defocusing nonlinear Schrödinger equation. We will discuss the existence of self-similar solutions and the stability near those solutions. During the talk, I will mention our work with Gonzalo Cao-Labora, Javier Gómez-Serrano, and Gigliola Staffilani on the first non-radial implosion result for those three equations. If there is time, I will also mention our work on self-similar solutions of a hydrodynamic equation that may formally arise as potential blow-up profiles of the focusing NLS equation.

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Date/Time : 22 Apr 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Khakim Egamberganov, National University of Singapore

We study the long-time behaviour of vorticity in three-dimensional axisymmetric Euler flows without swirl. This equation is globally well-posed when the initial vorticity satisfies some integrability conditions, but due to the presence of vortex stretching term, the vorticity and related quantities tend to grow in time. In this talk, we will discuss some improved upper and lower bounds for vorticity growth. This talk is based on joint work with Yao Yao.

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Date/Time : 18 Mar 2026 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Sam G. Krupa, École Normale Supérieure, Paris

For hyperbolic systems of conservation laws in 1-D, fundamental questions about uniqueness and blow up of weak solutions still remain even for the apparently “simple” systems of two conserved quantities such as isentropic Euler and the p-system. Similarly, in the multi-dimensional case, a longstanding open question has been the uniqueness of weak solutions with initial data corresponding to the compressible vortex sheet. We address all of these questions by using the lens of convex integration, a general method of constructing highly irregular and non-unique solutions to PDEs. Our proofs involve computer-assistance. This talk is based on joint work with László Székelyhidi, Jr.

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Date/Time : 18 Mar 2026 14:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Zihui Zhao, Johns Hopkins University

Free boundary problems are problems of solving an elliptic or parabolic PDE, but on a domain that is determined by the solution, and they arise naturally in the study of water waves and ice melting, etc. In this talk, we will focus on the regularity of the free boundaries for the Bernoulli problem, firstly studied by Alt and Caffarelli. More precisely, we study the behavior of such solutions near some well-known singularity models. In particular, our result implies that such singularity is not the typical behavior. This is joint work with Engelstein and Restrepo.

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Date/Time : 11 Mar 2026 14:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Nguyen Thi Van Anh, Hanoi National University of Education

The Allen–Cahn equation is a reaction–diffusion equation of mathematical physics which describes the process of phase separation in multi-component alloy systems, including order-disorder transitions. The relation between the approximate Allen–Cahn equations and mean curvature flow has been an ongoing investigation for several decades. For well-prepared initial conditions, Evans, Soner, and Souganidis proved the convergence to the viscosity solutions, Ilmanen showed convergence results to a Brakke flow, and many others. For a general initial conditions, Alfaro, Hilhorst and Matano carried out an analysis showing that the solution develops a steep transition layer within a specific time scale of parameters. In this talk, I will consider the Allen-Cahn equation under the initial condition that is no longer well-prepared and even discontinuous, in particular it is simply as a ‘signed function’ of a given set. The solvability of problems, the convergence to the mean curvature flows in both classical and weak sense are discussed. This is a part of an ongoing research project with Giovanni Bellettini.

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Date/Time : 11 Mar 2026 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Georg Weiss, University of Duisburg-Essen

While global solutions of the obstacle problem have recently been completely characterized (Eberle-Figalli-Weiss 2025), the class of global solutions to the Bernoulli problem is, by a result of M. Traizet (2013), much richer and closely related to minimal surfaces.

In this talk, we explain why a frequency formula of the type used in the obstacle problem is unlikely to be valid for the Bernoulli problem. We also present progress on a different approach—one that does not rely on complex analysis—which appears to be a promising alternative for studying global solutions of the Bernoulli problem.

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Date/Time : 25 Feb 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Chengyang Shao, Institut des Hautes Études Scientifiques

This is a mini-course with 3 lectures (1.5 hours each).   The time and venue are:
20 Feb (Friday) 3pm-4:30pm S16-03-06
23 Feb (Monday) 3pm-4:30pm S17-05-11
25 Feb (Wednesday) 3pm-4:30pm S17-05-11

This mini-course equips the audience with essential mathematical tools for the construction of stable and unstable invariant manifolds in hydrodynamical free boundary problems. In particular, it will be shown that the nonlinear instability for a moving capillary water jet follows precisely the Rayleigh-Plateau instability law. Being fully nonlinear and nonlocal in nature, such problems exhibit severe loss of derivatives and cannot be treated by standard methods. Therefore, it has remained largely open. Paradifferential calculus, however, provides a refined framework for analyzing nonlinearities, allowing them to be handled in a manner analogous to linear operators. It yields a unified framework addressing the construction of stable and unstable invariant manifolds for quasilinear PDEs or hydrodynamical free boundary problems.

The course consists of three lectures. The first lecture offers a concise introduction to classical partially hyperbolic dynamical systems and the Lyapunov-Perron method, followed by a review of its extensions to semilinear dispersive PDEs and Eulerian systems with fixed boundaries. The specific difficulties arising in quasilinear and free boundary problems are then discussed. The second lecture provides a brief introduction to the calculus of paradifferential operators, emphasizing its algebraic similarities with classical (pseudo)differential operators. In particular, it will be shown how this calculus applies to the analysis of Dirichlet-Neumann operator in free boundary problems. Finally, the third lecture demonstrates how these tools can be applied to construct stable and unstable invariant manifolds for hydrodynamical free boundary problems, thereby extending the classical Lyapunov-Perron method to this setting.

Some familiarity with the standard theory of hyperbolic PDEs, such as paraproduct decomposition and energy estimates, will be helpful. No prior background in dynamical systems is required.

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