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Colloquium. Academic Year 23-24

Thursdays 4:10pm

SC 5211

Colloquium Chair (2022-2023): Denis Osin


March 2: Chun Liu (Illinois Tech),

Host: Gieri Simonett


March 9: David Fisher (Rice University)

Host: Denis Osin

March 30: Matt Kennedy (University of Waterloo)

Host: Jesse Peterson

Title: The algebraic structure of operator algebras constructed from groups
Abstract: Since the work of von Neumann, the theory of operator algebras has been inextricably linked to the theory of groups. On the one hand, operator algebras constructed from groups provide an important source of examples and insight. On the other hand, many problems about the groups are most naturally studied within an operator-algebraic framework. In this talk, I will give an overview of some problems relating the structure of a group to the structure of a corresponding operator algebra, and describe some recent developments.


April 6: Romain Tessera (Institut de Mathematiques de Jussieu-Paris Rive Gauche)

Host: Denis Osin

Title: Embedding problems in geometric group theory

Abstract: Geometric group theory considers finitely generated groups as metric objects, and classically studies them up to quasi-isometries. This line of research has led to impressive classification results, notably for lattices in semi-simple Lie groups. Beyond quasi-isometries, another natural family of maps is formed by coarse embeddings. For instance, subgroup inclusion is a coarse embedding. These maps also arise in pseudo-Riemannian geometry: for example, Gromov observed in the eighties that the isometry group of a closed Lorentz (n+1)-manifold coarsely embeds into the real hyperbolic space of dimension n. In this talk, I will expose recent developments in the study of coarse embeddings, comprising the following result: an amenable group coarsely embeds in a hyperbolic group if and only if it is virtually nilpotent.


April 13: Thomas Koberda (University of Virginia)

Host: Spencer Dowdall

Title: Groups and discrete diffeomorphism invariants

Abstract: I will outline a research program whose aim is to produce discrete diffeomorphism invariants of manifolds in the form of finitely generated groups acting on manifolds with prescribed levels of regularity. I will discuss how this program has been mostly completed in dimension one, and give perspectives on higher dimensions.


April 20: Sinan Gunturk (NYU)

Host: Alex Powell

Title: Approximation with one-bit polynomials and one-bit neural networks

Abstract:  The effectiveness of neural networks has often been explained through the “universal approximation” theorems which state how well a given class of functions can be approximated by networks of a given size. Keeping in mind that modern networks can have an enormous number of real parameters, this talk will address the extreme problem of whether these parameters can be chosen from a set consisting of two values only, such as {+1,-1}. As part our solution to this problem, we will focus on an even more elementary problem in which we will investigate the approximation power of polynomials with ±1-coefficients. The key to our solution is the use of the Bernstein basis which, somewhat interestingly, behaves much more like a redundant frame.
Joint work with Weilin Li.


May 24, 3-4PM: Donatella Danielli, Arizona State University

Location: SC 1307

Host: Mark Ellingham

Title: “Obstacle problems for fractional powers of the Laplacian”

Abstract: In this talk we will discuss a sampler of obstacle-type problems associated with the fractional Laplacian $(-\Delta)^s$ , for $1 < s < 2$. Our goals are to establish regularity properties of the solution and to describe the structure of the free boundary. To this end, we combine classical techniques from potential theory and the calculus of variations with more modern methods, such as the localization of the operator and monotonicity formulas. This is joint work with A. Haj Ali (Arizona State University) and A. Petrosyan (Purdue University).

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