Date/Time : 08 Jan 2025 15:30
Venue : IMS executive seminar room (S17 Level 3)
Speaker : Jeremie Szeftel, Sorbonne Universite

I will introduce the celebrated black hole stability conjecture according to which the Kerr family of metrics are stable as solutions to the Einstein vacuum equations of general relativity. I will then discuss the history of this problem, including a recent work on the resolution of the black hole stability conjecture for small angular momentum.

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Date/Time : 08 Jan 2025 16:30
Venue : S17 #04-04 (SR3)
Speaker : Govind Menon, Brown University

We study the training dynamics of the Deep Linear Network (DLN) from the perspective of the geometric

theory of dynamical systems. Rigorous results by several authors are unified into a thermodynamic framework for deep learning. The analysis begins with a characterization of the invariant manifolds and Riemannian geometry in the DLN. This is followed by exact formulas for a Boltzmann entropy, as well as stochastic gradient descent of free energy using a Riemannian Langevin Equation. Several links between the DLN and other areas of mathematics are discussed, along with some open questions.

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Date/Time : 07 Jan 2025 16:00
Venue : S17 #05-12 (SR4)
Speaker : Chen Wan, Rutgers University—Newark

In this talk, I will discuss the geometric expansion of a local (twisted) trace formula for some special varieties. This generalizes the local (twisted) trace formula for reductive groups proved by Arthur and Waldspurger. By applying the trace formula, we prove a multiplicity formula for these spherical varieties. And I will also discuss some applications to the multiplicity of the Galois model and the unitary Shalika model. This is a joint work with Raphael Beuzart-Plessis.

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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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Date/Time : 27 Dec 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Jingbo Wan, Columbia University

In this talk, we discuss the rigid behaviour of contracting maps between positively curved manifolds via the method of geometric heat flows. We establish rigidity results for distance and area non-increasing maps between spheres, complex projective spaces and other positively curved manifolds under various curvature conditions. The methods rely on energy monotonicity for harmonic map heat flow, and the Strong Maximum Principle for Mean Curvature Flow or the coupled Mean Curvature Flow-Ricci Flow system. These results demonstrate and extend an interesting application of geometric heat flows to the homotopy classes of maps, pioneered by Prof. Mao-Pei Tsui and Prof. Mu-Tao Wang in 2004. This is based on joint works with Man-Chun Lee and Luen-Fai Tam.

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Date/Time : 26 Dec 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Sang-hyun Kim, Korea Institute for Advanced Study

Given a topological (C0) action of a group on a manifold, one can ask the best possible regularity (Cr) with which the group may act smoothly. Estimation of such regularities involve interplays among group theory (nilpotence, solvability and so forth), analysis (Hölder regularity) and dynamics (tracing orbits). In this talk, We prove that if a countable group does not contain a finitely generated subgroup of exponential growth, then every topological action of the group on a compact connected one-manifold can be blown-up to a C1 action. The proof is based on a functional characterisation of such groups. Joint work with Mikael de la Salle, Nicolas Matte Bon and Michele Triestino.

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Date/Time : 20 Dec 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Shouhong Wang, Indiana University

The first part of the talk is to introduce the general framework of phase transition dynamics for deterministic dissipative systems, with various applications to statistical physics and fluid flows. This is based on joint work with Dr. Tian Ma. In the second part, we study transitions in stochastic non-equilibrium systems. As we know, a central challenge in physics is to describe non-equilibrium systems driven by randomness. For deterministic systems, the center manifold theory has shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns, associated with genuine physical regimes. In presence of random fluctuations, the underlying reduction principle to the center manifold is seriously challenged due to large excursions caused by the noise. We present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds, on one hand, and energy estimates measuring the defect of parameterization of the high-modes, on the other. As a result, the approach enables us to predict, from reduced equations of the stochastic fluid problem, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation. This is joint work with Mickael Chekroun, Honghu Liu, and James McWilliams [JDE, 346(2023), 145-204].

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Date/Time : 18 Dec 2024 10:45
Venue : S17 #04-05 (SR2)
Speaker : Robin Zhang, MIT

In general, local converse theorems say that generic or cuspidal representations can be uniquely determined by their Rankin–Selberg gamma factors. After Jacquet and Langlands first proved such a statement for complex representations of local GL(2), an extensive series of works has established local converse theorems for complex representations of classical groups, some other local quasi-split reductive groups, and finite GL(n). A “naïve local converse theorem” fails for modular representations of finite GL(n) in characteristic ℓ > 0, but I will discuss how to fix this and prove a local converse theorem for finite GL(n) after defining a more general gamma factor with values in certain ℤ[1/p, ζp]-algebras. In the spirit of Henniart’s proof that the local Langlands correspondence for GL(n) is uniquely characterized by the compatibility of gamma factors, one may hope that this can give a “finite field local Langlands in families” determined by gamma factors. This is based on joint work with J. Bakeberg, M. Gerbelli-Gauthier, H. Goodson, A. Iyengar, and G. Moss.

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Date/Time : 18 Dec 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Jiepeng Fang, University of Hong Kong

By using Lusztig sheaves on the framed quiver, we give a categorical realization of the tensor product of integrable highest weight modules of the quantum group such that simple perverse sheaves in the category provide the canonical basis of the tensor product. As a consequence, we obtain the positivity of the canonical basis of the tensor product. This talk is based on a joint work with Yixin Lan.

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