Abstracts
Week of April 25, 2021
Group Actions and Dynamics | Basepoint-independent density of almost-primes in horospherical orbits in SL(3,Z)\SL(3,R) |
4:00pm -
Zoom
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Inspired by the work of Sarnak and Ubis [1] in SL(2,Z)\SL(2,R), we prove that almost-prime times (i.e. integer times having fewer than a fixed number of prime factors) in horospherical orbits of generic points in SL(3,Z)\SL(3,R) are dense in the whole space, where the number of prime factors allowed in the almost-primes is independent of the basepoint. This is in contrast to previous work [2] in which the number of prime factors depends on a Diophantine property of the basepoint. The proof involves a case-by-case analysis of the different ways in which a basepoint can fail the Diophantine property. If a basepoint fails to equidistribute rapidly in the whole space with respect to the continuous time flow, then there exists a sequence of nearby periodic orbits of increasing volume that approximate the original orbit up to larger and larger time scales, and which equidistribute in the whole space as the volume grows. Given an open set, one can find a large enough periodic orbit such that almost-primes of a fixed order in the periodic orbit land inside that set, and this property can then be transported to the nearby orbit of the original basepoint. This is joint work-in-progress with Manuel Luethi. [1] Sarnak, Peter, and Adrián Ubis. “The horocycle flow at prime times.” Journal de mathématiques pures et appliquées 103.2 (2015): 575-618. |
Geometry, Symmetry and Physics | Integrable systems and Hamiltonian flows in Calabi-Yau categories |
4:30pm -
https://yale.zoom.us/j/92811265790 (Password is the same as last semester)
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I will describe a general approach to constructing Hamiltonian |
Algebra and Number Theory Seminar | Duality theorem for p-adic general spin groups |
9:00am -
Zoom
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The duality theorem for p-adic groups aims to realize the dual of irreducible admissible representations on the space of the given representation itself. In this talk we will present a proof of the duality theorem for p-adic general spin groups by constructing a suitable duality involution on the group. |
Applied Mathematics | On the Implicit Bias of Dropout |
2:00pm -
Zoom Meeting ID: 97670014308
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Abstract: Dropout is a simple yet effective regularization technique that has been applied to various machine learning tasks, including linear classification, matrix factorization and deep learning. However, the theoretical properties of dropout as a regularizer remain quite elusive. This talk will present a theoretical analysis of dropout for single hidden-layer linear neural networks. We demonstrate that dropout is a stochastic gradient descent method for minimizing a certain regularized loss. We show that the regularizer induces solutions that are low-rank, in the sense of minimizing the number of neurons. We also show that the global optimum is balanced, in the sense that the product of the norms of incoming and outgoing weight vectors of all the hidden nodes equal. Finally, we provide a complete characterization of the optimization landscape induced by dropout. Bio: Rene Vidal is the Herschel Seder Professor of Biomedical Engineering and the Inaugural Director of the Mathematical Institute for Data Science at The Johns Hopkins University. He has secondary appointments in Computer Science, Electrical and Computer Engineering, and Mechanical Engineering. He is also a faculty member in the Center for Imaging Science (CIS), the Institute for Computational Medicine (ICM) and the Laboratory for Computational Sensing and Robotics (LCSR). Vidal’s research focuses on the development of theory and algorithms for the analysis of complex high-dimensional datasets such as images, videos, time-series and biomedical data. His current major research focus is understanding the mathematical foundations of deep learning and its applications in computer vision and biomedical data science. His lab has pioneered the development of methods for dimensionality reduction and clustering, such as Generalized Principal Component Analysis and Sparse Subspace Clustering, and their applications to face recognition, object recognition, motion segmentation and action recognition. His lab creates new technologies for a variety of biomedical applications, including detection, classification and tracking of blood cells in holographic images, classification of embryonic cardio-myocytes in optical images, and assessment of surgical skill in surgical videos. email tatianna.curtis@yale.edu for info |
Geometry & Topology | Typical Weil--Petersson geodesics. | 4:00pm - |
The Weil--Petersson (WP) metric is an incomplete Riemannian metric on the moduli space of finite type Riemann surfaces and the completion is the Deligne--Mumford compactification. Since WP geodesics that get to the boundary strata in finite time have zero Liouville measure, the WP flow (as a flow) makes measure-theoretic sense. The dynamical properties of the flow then become objects of interest. Recently, the WP flow has been shown to be ergodic for non-exceptional moduli spaces and exponentially mixing for exceptional moduli. In the work that we will describe, we begin the first steps in leveraging these properties to derive statistics along typical geodesics. Our focus will be cusp excursion statistics along typical geodesics which can be thought of as analogues of logarithm laws/ shrinking target problems in this incomplete setting. Some of this is joint work with Carlos Matheus. |