How are free entropy and classical entropy related?

Jennifer Pi

November 3rd, 2023 at 4pm - PS NS2 1201

Abstract

D. Voiculescu introduced two main notions of free entropy for a given tuple of self-adjoint operators $X$: the microstates free entropy $\chi(X)$ and the non-microstates free entropy $\chi^*(X)$. In this talk, we will introduce the notion of entropy from a classical viewpoint, and use this to give an idea of the free (non-commutative) analogs of entropy. We will conclude with a sketch of an elementary proof of the inequality $\chi(X) \leq \chi^*(X)$ (originally proved by Biane, Capitaine, and Guionnet). The proof leverages relationships between the free entropy of a tuple X and the classical entropy of appropriate matrix approximations to X. This is based upon a joint work with David Jekel.

About the Speaker

Jenny is a fifth year graduate student, who broadly works on problems related to the model theory of operator algebras and free probability. You can see her website https://sites.google.com/view/jpi314/

 What Can Mass be? 

Tin Yau "Whisky" Tsang

June 12th, 2023  -  1:30pm - 2:30pm - RH 306

Abstract

It is a continuation of last year's talk "Mass and Geometry". We will see the inspiration from geometry help define mass for a local object. 

About the Speaker

He is a Whisky.

 Affine Comonoids as Noncommutative Spectra 

So Nakamura

June 9th, 2023  -  3:30pm - 4:30pm - RH 340N

Abstract

The maximal spectrum of a commutative algebra (over a field) is the set of the maximal ideals equipped with the Zariski topology. This construction is functorial and every such algebra can be viewed as the ring of functions on the maximal spectrum. In this talk, we construct from a noncommutative algebra a comonoid object in the category of affine spaces that is equipped with a topology, and view the algebra as a ring of functions on the comonoid. We further see that this construction defines a fully-faithful functor (between some appropriate categories) that restricts to the maximal spectrum functor, and discuss some open questions we can ask.

About the Speaker

So is a 3rd year PhD student. His advisor is Manny Reyes. 

 The anisotropic Bernstein problem

Yang Yang

April 13th, 2023  -  5:00pm - 6:00pm - RH 306

Abstract

The Bernstein problem asks whether entire minimal graphs in R^{n+1} are necessarily hyperplanes. It is known through spectacular work of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De Giorgi-Giusti that the answer is positive if and only if n < 8. The anisotropic Bernstein problem asks the same question about minimizers of parametric elliptic functionals, which are natural generalizations of the area functional that both arise in many applications, and offer important technical challenges. We will discuss the recent solution of this problem (the answer is positive if and only if n < 4). 

About the Speaker

Yang is currently a 6th year PhD advised by Connor Mooney. His research interest is Partial Differential Equations and Geometric Analysis. Yang is particularly interested in questions concerning regularity and classification of global solutions to geometric variational problems. 

About this talk

This is joint work with C. Mooney.

 A Model Theoretic Free Group Factor Alternative  

Jennifer Pi

March 8th, 2023  -  4:30pm - 5:30pm - RH 306

Abstract

A free group on n generators is a group generated by n elements with no relations. While it is known that the free groups are non-isomorphic for different numbers of generators, it is still open whether the free group factors (von Neumann algebras created using the free groups) with distinct numbers of generators are isomorphic or not. The best-known result currently (done by Dykema and Radulescu independently in the early 90's) establishes that they are all isomorphic or pairwise non-isomorphic. In this talk, we will discuss some ongoing work toward a model-theoretic version of this dichotomy. 

About the Speaker

Jenny is a fourth-year graduate student studying jointly under Isaac Goldbring and Mike Cranston. Her research interests include model theory of operator algebras, free probability and free entropy, and (tentatively) representation theory and geometric group theory.

About this talk

We assume no knowledge of model theory or von Neumann algebras.