Week of October 17, 2021

October 18, 2021
Geometry, Symmetry and Physics Derived Satake equivalence for Godement-Jacquet monoids 4:30pm -

Abstract: We prove an equivalence between the equivariant derived category of constructible sheaves on the loop space of n-by-n matrices and the category of perfect complexes on a certain formal dg algebra. This equivalence is compatible with actions of the derived Satake category. I will explain how our result is related to conjectural dualities in the Langlands program and in physics. The proof involves certain invariant theoretic properties of a certain "dual" cotangent bundle. This is joint work with Tsao-Hsien Chen (in preparation).

Zoom link: https://yale.zoom.us/j/99305994163, contact the organizers (Gurbir Dhillon and Junliang Shen) for the passcode.

October 19, 2021
Geometry & Topology On the Weil-Petersson gradient flow of renormalized volume on a Bers slice 4:15pm -
LOM 214

We show that the flow on a Bers slice given by the Weil-Petersson gradient vector field of renormalized volume is globally attracting to its fuchsian basepoint. This is based on joint work with Martin Bridgeman and Ken Bromberg.

Algebra and Number Theory Seminar Generic Langlands Functoriality Conjecture for SO*(2n) in positive characteristic 4:30pm -

The Langlands functoriality conjecture is one of the far-reaching conjectures of the Langlands program.

Over number fields, Cogdell, Kim, Piatetski-Shapiro, and Shahidi prove this conjecture for globally generic cuspidal automorphic representations from the split classical groups, unitary groups or quasi-split special orthogonal groups to the general linear groups. Lomelí extends this result to split classical groups and unitary groups in positive characteristics.

In this talk, we are going to study Langlands functoriality conjecture for globally generic cuspidal automorphic representations of quasi-split non-split even special orthogonal groups in positive characteristics by means of the converse theorem combine with the Langlands-Shahidi method.

October 20, 2021
Applied Mathematics Probabilistic Variable Selection 2:30pm -

The computational resource growth in natural science motivates the use of machine learning for automated scientific discovery. However, unstructured empirical datasets are often high dimensional, unlabeled, and imbalanced. Therefore, discarding irrelevant (i.e., noisy and information-poor) features is essential for the automated discovery of governing parameters in scientific environments. We present Gaussian Stochastic Gates (STG), which rely on a probabilistic relaxation of the L0 norm of the number of selected features to address this challenge. By applying the Stochastic Gates to a neural network’s input layer, we derive a flexible, fully differentiable model that simultaneously identifies the most relevant features and learns complex nonlinear models. The STG neural network outperforms the state-of-the-art feature selection methods, both in terms of predictive power and its ability to correctly identify the correct subset of informative features. The model was successfully applied for critical biological tasks such as COX proportional hazards model and differential expression analysis on HIV and Melanoma patients. Next, using a linear model, we provide a theoretical basis for optimizing the STG objective using small batches (i.e., SGD). In particular, we present an approximation bound for estimating an unknown signal based on noisy observations. Finally, we develop an extension of the STG model for unsupervised feature selection. The new model is trained to select highly correlated features with the leading eigenvectors of a gated graph Laplacian. The gating mechanism allows us to re-evaluate the Laplacian for different subsets of features and unmask informative structures buried by nuisance features. I will demonstrate that the proposed approach outperforms several unsupervised feature selection baselines.

October 21, 2021
Analysis Localization and Cantor spectrum for quasiperiodic discrete Schrödinger operators with asymmetric, smooth, cosine-like sampling functions 2:15pm -


A discrete Schrödinger operator $H_V = \varepsilon\Delta + V$ on $\ell^2(\mathbb{Z})$ is called Anderson localized if it exhibits a basis of exponentially decaying eigenvectors. If $V_n$ is sampled from a potential function by Diophantine rotations on the one-dimensional torus, $H_V$ is known to be almost-surely Anderson localized for sufficiently small $\varepsilon$ if the potential is either analytic, or cosine-like and symmetric. In this talk, we discuss a new perturbative proof of almost-sure localization for Schrödinger operators with potential sampled from any $C^2$-smooth Morse function with two monotonicity intervals along a Diophantine rotation orbit on the circle. This is a joint work with Tom VandenBoom.

October 22, 2021
Geometric Analysis and Application Rotational symmetry of ancient solutions of Ricci flow in higher dimensions 2:00pm -


In this talk, we will discuss symmetry inheritance in the Ricci flow in higher dimensions under suitable curvature positivity assumptions. After some background, we will cover the three main ingredients of the proof: a linear analysis on the cylinder, the important neck improvement theorem, and the contradiction argument for cap symmetry (closely following the argument in 3D). Time permitting, we will compare the argument to its counterpart for mean curvature flow and mention some potential applications to the mean curvature flow in higher codimension. This is joint work with Simon Brendle.