Colloquia and Seminars
Outer Automorphisms of Free Groups and Free Products of Groups
Speaker: Rylee Alanza Lyman (Rutgers University–Newark)
Tuesday, May 3, 2022, at 11:00 a.m. in ES-245
Abstract: One fruitful way of understanding a mathematical object is to understand its group of symmetries. This approach can even be applied to groups themselves, leading to the study of outer automorphisms groups. Free groups are in some sense very simple groups to understand, but their outer automorphisms are still far from completely understood. Likewise, given a family of groups, their free product is the algebraically minimal way to combine them, but outer automorphisms of free products of groups are just beginning to be understood. In this talk, we will learn to think about free groups and free products topologically by means of graphs and trees. We will think of their outer automorphisms as transformations called homotopy equivalences of graphs and trees, culminating in my construction of particularly nice homotopy equivalences called relative train track maps and CTs for outer automorphisms of free products.
Ninth Annual Maheshwari Colloquium
Speaker: Mihai Putinar (University of California at Santa Barbara)
Friday, April 15, 2022, at 4:00 p.m. on Zoom
Mihai Putinar is Professor in the Mathematics Department at the University of California at Santa Barbara. Transylvanian by origin, he is now an international servant and ambassador of mathematics. With prior contributions to complex analytic geometry, real algebra, and moment problems, his recent works are related to positivity preservers and the structure of non-selfadjoint operators, mainly touching spectral theory, inverse problems, and approximation theory. He is the author of four books and more than two hundred research articles. He was a Humboldt Fellow and a Gambrinus Fellow. Over the course of his career, he has been awarded prizes including the Simion Stoilow Prize of the Romanian Academy (1987) and the Romanian National Order of Merit with the rank of Knight (2011).
Visit the Maheshwari Colloquium web page.
Orbital Integrals, Cyclic Cocycle, and Index Theory
Speaker: Xiang Tang (Washington University in St. Louis)
Friday, October 15, 2021
Abstract: Orbital integral is an integral transform on functions of a Lie group. It is an important tool in representation theory. In this talk, we will introduce a generalization of orbital integral and apply it to study invariant elliptic operators.
Deep Learning for the Discovery of Parsimonious Physics Models
Speaker: J. Nathan Kutz (University of Washington)
Friday, October 29, 2021
A major challenge in the study of dynamical systems is that of model discovery: turning data into reduced order models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SINDy to scale efficiently to problems with multiple time scales, noise and parametric dependencies. For systems with incomplete observations, we show that the Hankel alternative view of Koopman (HAVOK) method, based on time-delay embedding coordinates and the dynamic mode decomposition, can be used to obtain linear models and Koopman invariant measurement systems that nearly perfectly captures the dynamics of nonlinear quasiperiodic systems. Neural networks are used in targeted ways to aid in the model reduction process. Together, these approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.
Graph Rules for Inhibitory Network Dynamics
Speaker: Carina Curto (Pennsylvania State University)
Friday, November 5, 2021
Abstract: Many networks in the nervous system possess an abundance of inhibition, which serves to shape and stabilize neural dynamics. The neurons in such networks exhibit intricate patterns of connectivity, whose structure controls the allowed patterns of neural activity. In this work, we examine inhibitory threshold-linear networks whose dynamics are dictated by an underlying directed graph. We develop a set of parameter-independent graph rules that enable us to predict features of the dynamics from properties of the graph. These rules provide a direct link between the structure and function of recurrent networks, and yield new insights into how connectivity may shape dynamics in real neural circuits. We will illustrate this with some applications to central pattern generator circuits and other examples of neural computation.
Harmonic Persistent Homology
Speaker: Saugata Basu (Purdue University)
Friday, December 3, 2021
Abstract: I will introduce harmonic persistent homology spaces for filtrations of finite simplicial complexes. As a result, it is possible to associate concrete subspaces of cycles to each bar of the barcode of the filtration. I will discuss the stability of the harmonic persistent homology subspaces under small perturbations of functions defining them. Finally, I will relate the notion of "essential simplices" introduced in an earlier work to identify simplices that play a significant role in the birth of a bar, with that of harmonic persistent homology. We prove that the harmonic representatives of simple bars maximize the "relative essential content" amongst all representatives of the bar, where the relative essential content is the weight a particular cycle puts on the set of essential simplices. The talk is based on the joint work with Nathanael Cox.
Algebra/Topology Seminar (Thursdays at 3:00pm)
Analysis and Data Science Seminar (Tuesdays at 3:00pm)
Applied Topology in Albany (ATiA) Seminar (Fridays at 11:30am)