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 : 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 : 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 : 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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Date/Time : 24 Jan 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Wang Kelei, Wuhan University

Since the seminal work of Alt-Caffarelli in 1981, the one phase free boundary problem has been studied by many people. To study the regularity and singularity of free boundaries, blow up analysis is a standard method. It turns out for this free boundary problem, the nondegeneracy condition is crucial for the application of this method. Although this condition holds for energy minimizers, it’s not true for general solutions. In this talk, I will discuss a proof of the nondegeneracy for stable solutions. This is based on a joint work with N. Kamburov.

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Date/Time : 18 Jan 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Ruixiang Zhang, University of California, Berkeley

The Fourier restriction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space mapping properties of certain oscillatory integral operators. They both are central in harmonic analysis, are open in dimensions $\geq 3$, and notably have the same conjectured exponents. In the 1970s, H\”{o}rmander asked if a more general class of operators (known as H\”{o}rmander type operators) all satisfy the same $L^p$-boundedness as in the above two conjectures. A positive answer to H\”{o}rmander’s question would resolve the above two conjectures and have more applications such as in the manifold setting. Unfortunately H\”{o}rmander’s question is known to fail in all dimensions $\geq 3$ by the work of Bourgain and many others. It continues to fail in all dimensions $\geq 3$ even if one adds a “positive curvature” assumption which one does have in restriction and Bochner-Riesz settings. Bourgain showed that in dimension $3$ one always has the failure unless a derivative condition is satisfied everywhere. Joint with Shaoming Guo and Hong Wang, we generalize this condition to arbitrary dimension and call it “Bourgain’s condition”. We unify Fourier restriction and Bochner-Riesz by conjecturing that any H\”{o}rmander type operator satisfying Bourgain’s condition should have the same $L^p$-boundedness as in those two conjectures. As evidences, we prove that the failure of Bourgain’s condition immediately implies the failure of such an $L^p$-boundedness in every dimension. We also prove that current techniques on the two conjectures apply equally well in our conjecture and make some progress on our conjecture that consequently improves the two conjectures in higher dimensions. I will talk about some history and some interesting components in our proof.

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Date/Time : 29 Nov 2023 15:00
Venue : S17 #06-11 (SR6)
Speaker : Dawei Shen, Sorbonne University

The global stability of Minkowski spacetime has been proven in the celebrated work of Christodoulou-Klainerman in 1993. In 2007, Bieri has extended the result of Christodoulou-Klainerman under lower decay and regularity assumptions on the initial data. In this talk, we will introduce a recent work, which extends the result of Bieri to minimal decay assumptions.

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Date/Time : 29 Nov 2023 14:00
Venue : S17 #06-11 (SR6)
Speaker : Jaemin Park, University of Basel

In this talk, we study stationary solutions to the 2D incompressible Euler equations in the whole plane. It is well-known that any radial vorticity is stationary. For compactly supported vorticity, it is more difficult to see whether a stationary solution has to be radial. In the case where the vorticity is non-negative, it has been shown that any stationary solution has be radial. By allowing the vorticity to change the sign, we prove that there exist non-radial stationary patch-type solutions. We construct patch-type solutions whose kinetic energy is infinite or finite. For the finite energy case, it turns out that a construction of a stationary solution with compactly supported velocity is possible. This is based on the joint work with Javier Gomez-Serrano and Jia Shi.

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