Date/Time : 09 Jan 2025 14:00
Venue : S17 #04-06 (SR1)
Speaker : Jian Ding, Peking University

I will describe various percolation models in statistical physics, with emphasis on phase transition, critical phenomenon as well as disorder effects. In particular, I will start my talk with Bernoulli percolation, and then present a few important examples of correlated percolation models. While I try to be a “messenger” communicating progress accomplished in the community as a whole, the talk is inevitably biased by my personal interests (for instance, I will spend some time surveying recent progress on the random field Ising model and Gaussian free field related percolation models).

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Date/Time : 09 Jan 2025 16:00
Venue : S17 #04-06 (SR1)
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 : 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 : 08 Jan 2025 14:00
Venue : IMS executive seminar room (S17 Level 3)
Speaker : Wendelin Werner (Recipient of the Fields Medal in 2006), University of Cambridge

Physical theories are often described in terms of fields and properties of their correlation functions. I will discuss several older and recent instances, where the understanding of related random underlying geometric objects and their properties turned out to be interesting and fruitful.

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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 : 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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