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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Date/Time : 15 Jan 2025 15:30
Venue : S17 #04-04 (SR3)
Speaker : Ye Zikai, Chinese University of Hong Kong

Yang-Mills-Higgs equation plays an important role in modern physics and geometry. In this talk, I will introduce the existence and stability of certain solutions to the U(1)-Yang-Mills-Higgs equation in R^4. I will begin with the standard vortex solutions in R^2 and how Liu-Ma-Wei-Wu used them to construct a solution in R^4 that concentrates near the Arrezo-Pacard surface by the Lyapunov reduction method. Then I will show this solution is stable due to the stability of the standard vortex solution and the Arrezo-Pacard surface as a holomorphic curve. Finally, I will mention several ongoing projects on this topic.

This is the joint work with Yong Liu and Juncheng Wei.

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Date/Time : 27 Dec 2024 15:00
Venue : S17 #05-12 (SR4)
Speaker : Dawei Shen, Columbia University

Kerr stability for small angular momentum has been proved by Klainerman-Szeftel, Giorgi-Klainerman-Szeftel and Shen in a series of works. Some of the most basic conclusions, concerning various physical quantities on the future null infinity are derived therein. Further important conclusions were later derived by An-He-Shen and Chen-Klainerman. In this talk, I will present the existence and uniqueness of a canonical foliation on the future null infinity for which the energy, linear momentum, center of mass and angular momentum are well defined and satisfy the expected physical laws of gravitational radiation. The rigid character of this foliation eliminates the usual ambiguities related to these quantities in the physics literature. This is based on a joint work with Sergiu Klainerman and Jingbo Wan.

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