Week of December 12, 2021

December 13, 2021
Colloquium The Fontaine-Mazur conjecture in dimension two Lue Pan - Princeton 3:00pm -
Zoom
December 15, 2021
Applied Mathematics New Methods for Low-Rank Matrix Denoising WIlliam Leeb - University of Minnesota 2:30pm -
https://yale.zoom.us/j/97458245891
Colloquium L-functions, functoriality, and quantization Yiannis Sakellaridis - Johns Hopkins 4:15pm -
December 17, 2021
Geometric Analysis and Application Topological rigidity of the first Betti number Shaosai (Arthur) Huang - University of California at Berkeley 2:00pm -

Abstracts

Week of December 12, 2021

December 13, 2021
Colloquium The Fontaine-Mazur conjecture in dimension two 3:00pm -
Zoom

The Fontaine-Mazur conjecture predicts which two-dimensional p-adic representations of the absolute Galois group of Q arise from modular forms. In this talk, I will explain this conjecture by some concrete examples and report some recent progress. 
December 15, 2021
Applied Mathematics New Methods for Low-Rank Matrix Denoising 2:30pm -
https://yale.zoom.us/j/97458245891

Abstract: In this talk, I will describe a new class of methods for estimating a low-rank matrix from a noisy observed matrix, where the error is measured by a type of weighted loss function. Such loss functions arise naturally in problems such as submatrix denoising, heteroscedastic noise filtering, and estimation with missing data. I will introduce a family of spectral denoisers, which preserve the left and right singular subspaces of the observed matrix. I will also show how denoising with weighted loss yields a new approach to unweighted denoising that is asymptotically at least as good as singular value shrinkage, and can perform better when the matrix is heterogeneous. I will demonstrate the behavior of the method through numerical simulations.
Colloquium L-functions, functoriality, and quantization 4:15pm -

Abstract:

L-functions of number fields, elliptic curves, and other objects of diophantine geometry, are ubiquitous in number theory, but their study almost always relies on the Langlands correspondence, which predicts that they encode “eigenfrequencies” of arithmetic manifolds, and realizes them as “period integrals” of automorphic forms on such manifolds (e.g., of modular forms).

After introducing these notions, I will explain how the idea of quantization of symplectic manifolds can provide a symmetric interpretation of the duality between L-functions and periods (based on ongoing joint work with D. Ben-Zvi and A. Venkatesh), and will present evidence for its relevance to the Langlands functoriality conjecture (“different arithmetic drums share the same eigenfrequencies”), based on recent work of mine on the rank-1 case.
December 17, 2021
Geometric Analysis and Application Topological rigidity of the first Betti number 2:00pm -

Abstract:

The infranil fiber bundle is a typical structure appeared in the collapsing geometry with bounded sectional curvature. In this talk, I will discuss a topological condition on the first Betti numbers that guarantees a torus fiber bundle structure (a special type of infranil fiber bundle) for collapsing manifolds with only Ricci curvature bounded below. A major technique applied here is smoothing by Ricci flows. This is a joint work with Bing Wang.