Date/Time : 04 Aug 2026 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Giovanni Brigati, Institute of Science and Technology, Austria

In this talk, I am proposing a gradient-flow interpretation for a class of non-linear kinetic Fokker—Planck equation (among which the linear case of the Vlasov—Fokker—Planck). The challenge lies in the fact that such dynamics are driven by an interplay of conservative and dissipative effects. Namely, free transport, and diffusion in the velocity variables only. Thus, we rely on the geometry of “kinetic optimal transport”, which is based on Newton’s equations. In this framework, regular curves of measures can be rewritten as solutions to Vlasov’s equations, for a certain force field.

The first main result of the talk is a chain rule for a family of free energy functionals along Vlasov’s equations, which allows us to compute the slope of the energy. Curves of maximal slope correspond to solutions to the non-linear kinetic Fokker—Planck equation. I will then discuss convergence for JKO-like schemes, and further analytical properties of the non-linear kinetic equations, together with a few open questions. My presentation is based on joint works with Guillaume Carlier, Jean Dolbeault, Jan Maas, and Filippo Quattrocchi.

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Date/Time : 28 Jul 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Zhongkai Tao, Institut des Hautes Études Scientifiques

The initial data of the vacuum Einstein equations on a spacelike hypersurface satisfy the Einstein constraint equations. It is an underdetermined PDE system. We exploit the underdeterminedness to find nontrivial solutions of the Einstein constraint equations with interesting properties. A key ingredient is the construction of Bogovskii-type operators for underdetermined PDEs satisfying a robust symbol condition. The Bogovskii-type operators have favorable support and regularizing properties, allowing us to manipulate the solutions flexibly. This is based on joint work (partly in progress) with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Sung-Jin Oh (UC Berkeley)

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Date/Time : 28 Jul 2026 16:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Chengyang Shao, Institut des Hautes Études Scientifiques

In this talk, I will present a mathematical proof for the existence of time-periodic oscillating capillary droplet. The major difficulty is that the non-flat underlying geometry makes toric Fourier techniques unavailable, while the quasilinear nature still requires delicate spectral analysis. The analysis starts with paraproduct on compact Lie groups based on invariant Littlewood-Paley decomposition, followed by a toolset of explicit formulas for paradifferential operators. In case of zonal symmetry on the 3-sphere, an effectively computable symbolic calculus is available. It will then be shown how this paradifferential toolset is used to construct time-periodic solution to the spherical capillary water waves equation, giving mathematical justification of Rayleigh’s formula for oscillating capillary droplet. The paradifferential method not only incorporates the underlying geometry and analysis, but also effectively balances the loss of regularity caused by small denominators, allowing a “fixed point” style proof instead of Nash-Moser/KAM iteration. The talk is based on a working paper co-authored with Haocheng Yang.

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Date/Time : 23 Jul 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Xuantao Chen, Sorbonne Universite

We revisit the problem of solving the Einstein constraint equations in vacuum by a new method, which allows us to prescribe four scalar quantities, representing the full dynamical degrees of freedom of the constraint system. We show that once appropriate gauge conditions have been chosen and four scalars freely specified (modulo $\ell\leq 1$ modes), we can rewrite the constraint equations as a well-posed system of coupled transport and elliptic equations on two-spheres. Our method can be applied to construct various types of initial data, including data with arbitrarily fast decay, borderline decay, and short-pulse type data. This is joint work with Sergiu Klainerman.

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Date/Time : 08 Jul 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Wen Qi Zhang, Australian National University

It is known that $L^1$ Sobolev and Stein-Weiss inequalities follow a slightly different pattern to their $L^p$ counterparts. For instance, they serve as weaker replacement inequalities for the failure of $L^1$ inequalities between differential operators. Recent work by P. De Napoli and T. Picon has shown that the canceling/cocanceling framework introduced by J. Van Schaftingen can be used to extend the $L^1$ Stein-Weiss inequalities beyond their expected range. Inspired by these ideas, we investigate if the cocanceling constraint can be weakened in a subcritical setting, and we also explore how one may further extend these inequalities to a critical case.

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Date/Time : 07 Jul 2026 11:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Weiren Zhao , NYU Abu Dhabi

In this talk, I will talk about the long-time behavior of solutions to the two-dimensional Navier–Stokes equations with initial data evolving under the influence of the planar Couette flow. We focus on general perturbations, which may be large and of low regularity, and show that the vorticity asymptotically approaches a constant multiple of the fundamental solution of the corresponding linearized vorticity equation after a long-time evolution determined by the relative Reynolds number.

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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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