Date/Time : 23 Apr 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Luca Spolaor, University of California San Diego

I will discuss the boundary regularity for minimizers of the so-called Alt-Caffarelli problem and how it is related to a boundary unique continuation problem. This is joint work with L. Ferreri (SNS) and Bozhidar Velichkov (Pisa).

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Date/Time : 16 Apr 2025 16:00
Venue : S17 #04-04 (SR3)
Speaker : Weijun Xu, Peking University

We consider two types of stochastic perturbations to the one dimensional Allen-Cahn equation, one with spacetime white noise, and the other with more singular noise. In both models, the interface location, when suitably rescaled, converges to a diffusion process on the real line. Cancellations play important roles in both models, but they happen in very different manners.

In the first model, we construct “correctors” that are designed to cancel potential divergences. We thus recover classical results by Funaki (1995) and Brassesco-Butta-De Masi-Presutti (1998), but also shed some new thoughts. In the second model, the cancellation happens somewhat miraculously, and the result seems to be new. Based on joint works with Wenhao Zhao (EPFL) and Shuhan Zhou (PKU).

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Date/Time : 16 Apr 2025 15:00
Venue : S17 #05-12 (SR4)
Speaker : Mu-Tao Wang, Columbia University

Two black holes rotate about each other and eventually merge into a single black hole. How does one measure the angular momentum carried away by gravitational radiation during this merger? This has been a subtle issue since the 1960’s due to the existence of “supertranslation ambiguity”: the angular momentums recorded by two observers of the same system may not be the same.

In this talk, I shall describe how the mathematical theory of quasilocal mass and optimal isometric embedding identifies a new definition of angular momentum that is free of any supertranslation ambiguity. In addition, some recent development of the cross-section continuity of the angular momentum definition will also be discussed. This is based on joint work with Po-Ning Chen, Jordan Keller, Daniel Paraizo, Robert Wald, Ye-Kai Wang, and Shing-Tung Yau.

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Date/Time : 26 Mar 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Kim Sunghan, Uppsala University

Constraint maps are critical points to the Dirichlet energy subject to an image constraint. Namely, the constraint restricts these maps to lie outside a given (smooth, open-connected) set in the target space, which can then be thought of as an obstacle. The target space is multi-dimensional, making this problem a natural extension of the obstacle problem into the vectorial setting. The presence of the obstacle naturally induces free boundary, which is the interface between the set where the image of the maps lies away from the obstacle and the set where their image lies on the obstacle. Yet, constraint maps also develop singularities, such as discontinuous points and branch points which have been the main subjects in the theory of harmonic maps and minimal surfaces, respectively. In this talk, I will discuss the recent breakthroughs on the tantalizing interplay between the free boundary and the mapping singularities for energy-minimizing constraint maps, based on a series of my collaboration with Alessio Figalli, André Guerra (ETH), and Henrik Shahgholian (KTH).

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Date/Time : 17 Mar 2025 16:00
Venue : S16 #04-35
Speaker : Jiajie Chen , New York University

While it is known that a pre-shock or implosion singularity can form in finite time for the compressible Euler equations from smooth initial data, whether the vorticity blows up in finite time has remained an open problem. In this talk, we will explore the Euler equations with axisymmetry. Using (nearly) axisymmetric flow, we construct vorticity blowup in the compressible Euler equations in R^d, d >=2 with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity and is accompanied by a (nearly) axisymmetric implosion, in which the swirl velocity exhibits full stability as opposed to finite co-dimension stability.

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Date/Time : 17 Feb 2025 15:00
Venue : S17 #05-11 (SR5)
Speaker : Li Mengni, Southeast University

In this talk we focus on the inverse scattering problem of nonlinear Alfvén waves governed by the three dimensional ideal incompressible magnetohydrodynamics (MHD) system. By bridging together geometric methods and weighted energy estimates, we establish a set of scattering isomorphisms to substantially strengthen our previous rigidity results. This answer is consistent with the physical intuition that Alfvén waves behave exactly in the same manner as their scattering fields that are detected by faraway observers. The novelty of the present work is twofold: for one thing, the relationship between Alfvén waves emanating from the plasma and their scattering fields at infinities is explored to the best; for another thing, the null structure inherent in MHD equations is thoroughly examined, especially when we estimate the pressure term.

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Date/Time : 14 Feb 2025 15:00
Venue : S16 #03-06
Speaker : Xiong Jingang, Beijing Normal University

I will talk about the extinction dynamics of nonnegative solutions to the fast diffusion equation within bounded, smooth domains subject to the Dirichlet zero boundary condition. Initially, I will provide a concise overview of the comprehensive understanding achieved for the subcritical regime of this equation. Subsequently, I will shift my focus to the critical case, which remains an active area of investigation. Under the two-bubble energy threshold assumption on the initial data, we prove the dichotomy that every solution converges uniformly, in terms of relative error, to either a steady state or a blowing-up bubble. This is joint with Tianling Jin at HKUST.

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Date/Time : 24 Jan 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Zhihan Wang, University of Chicago

I’ll discuss an upcoming work joint with Otis Chodosh, Christos Mantoulidis and Felix Schulze, in which we prove generic regularity of area-minimizing hypersurfaces in ambient dimension 11, extending the previous result of my co-authors in dimension 9 & 10.

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Date/Time : 17 Jan 2025 15:30
Venue : S17 #04-04 (SR3)
Speaker : Tang Lan, Central China Normal University

We mainly study regularity theory for the Monge-Ampère equations. Firstly, we establish the optimal global C^(1,α) regularity results of generalized solutions to the degenerate Monge-Ampère equations; secondly, we give the free boundary regularity for a class of obstacle problem.

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