Date/Time : 10 Apr 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Maciej Dunajski, University of Cambridge

We prove that the intrinsic Riemannian geometry of compact cross-sections of any Einstein extremal horizon must admit a Killing vector field. This establishes the rigidity theorem for the extremal Kerr black hole horizon. This extremal horizon is a special case of a quasi-Einstein structure. We shall discuss other examples of quasi-Einstein structures on compact surfaces, including a global result relevant in projective metrizability.

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Date/Time : 27 Mar 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Shi-Zhuo Looi, California Institute of Technology

We present a framework and algorithm for deriving the sharp asymptotics of linear and non-linear wave equations on asymptotically flat spacetimes with even spatial dimensions, including non-stationary spacetimes without spherical symmetry assumptions.

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Date/Time : 20 Mar 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Wang Shu, Beijing University of Technology

In this talk we discuss quasi-neutral limit problem of Planck-Nernst-Poisson-Navier-Stokes equations for electro-hydrodynamics and review the known main progress on rigorously establishing quasi-neutrality theory from the view of mathematics. Quasi-neutrality is one basic assumption in physics such as semiconductors and plasma, which is firstly proposed by W.Van Roosbroech in Bell System Tech. J., 1950. Firstly, we introduce some multi-scaling structure stability theory on quasi-neutral limit and boundary layer, initial layer and mixed layer problem for drift-diffusion models in semiconductors. Then, we give some recent progresses on well-posedness theory and quasi-neutral limit problem on Planck-Nernst-Poisson-Navier-Stokes equations for electro-hydrodynamics.

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Date/Time : 13 Mar 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Willie Wong, Michigan State University

Most of us will agree that solving ordinary differential equations is easier than solving partial differential equations. As a consequence, many predictions of physical theories are obtained through considering regimes where the equations of motion reduce to a finite dimensional dynamical system. A specific example is in our understanding of the early universe, where the prevalent paradigm is based in part on cosmological big bang solutions in general relativity. This cosmological ansatz where the universe is assumed to be spatially homogeneous is often justified physically by the “working assumption” that when viewed from the largest scales, the inhomogeneities (stars, planets, galaxies) average out, and the matter content of the universe can be approximated by a compressible fluid. In a joint work with Shih-Fang Yeh, we probed whether this “working assumption” is justified, from a mathematical point of view. Our results show that on the cosmological timescale, certain cosmological big bang solutions are susceptible to instabilities generated through nonlinear self-interactions of the constituent inhomogeneous matter. The goal of this talk is to present the mathematical context of this result and briefly describe the instability mechanism involved, focusing on the relevance of the conformal (or causal) geometry of the big bang solution.

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Date/Time : 13 Mar 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Taoran He, National University of Singapore

In this talk, we will report and propose a new angular momentum memory effect of gravitational waves along null infinity based on recent progress in proving stability of Minkowski spacetime with minimal decays and nonlinear stability of Kerr black holes with small angular momentum. This accompanies Christodoulou’s nonlinear displacement memory effect and the spin memory effect. The connections and differences to these effects are also addressed. This is a joint work with Xinliang An and Dawei Shen.

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Date/Time : 21 Feb 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Chengyang Shao, University of Chicago

The speaker aims to report some advances in the study of irrotational oscillation of a water droplet under zero gravity. The governing physical laws, resembling the well-studied capillary water waves equation, are converted to a quasilinear dispersive para-differential system defined on the 2-sphere. Regarding this conversion, a coordinate-independent, global para-differential calculus defined on compact Lie groups and homogeneous spaces is developed as a toolbox. After discussing Cauchy theory under this novel formalism, the speaker will propose several problems concerning existence of periodic solutions, normal form reduction and generic lifespan estimates. It is pointed out that these problems are closely related to certain Diophantine equations.

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Date/Time : 21 Feb 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Zhaojie Yang, Fudan University

In this talk, I will present a backward uniqueness result of bounded mild solutions of 3D Navier-Stokes Equations in the whole space with non-trivial final data. A direct consequence is that a solution must be axi-symmetric in [0, T] if it is so at time T. The proof is based on a new weighted estimate which enables to treat terms involving Calderon-Zygmund operators. The new weighted estimate is expected to have certain applications in control theory when classical Carleman-type inequality is not applicable. This is a joint work with Prof. Zhen Lei and Dr. Cheng Yuan.

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Date/Time : 07 Feb 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Marco Badran, ETH Zurich

Given a closed manifold $N$ and a compact, non-degenerate minimal submanifold $M$ of codimension 2 we construct a collection of solutions $U_\varepsilon$ to the magnetic Ginzburg-Landau equations such that $U_\varepsilon$ concentrates around $M$ as $\varepsilon\to 0$, establishing a codimension two analogous of the celebrated result by Pacard and Ritoré in the Allen-Cahn setting. The construction is carried out via an asymptotic gluing technique, yielding precise asymptotic expansion of the concentration profile.

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Date/Time : 31 Jan 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Han Qing, University of Notre Dame

A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this talk, we study whether these metrics have negative Ricci curvatures. We present two classes of results. First, we prove that the Ricci curvature is indeed negative for the complete conformal metrics on convex domains. Second, we demonstrate the Ricci curvature splits in sign if a component of the boundary of the domain is close to a lower dimensional set.

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