Schedule of Talks, Abstracts
Schedule of Talks
All talks will be held in Stevenson Center 1432. All breaks will be held in Stevenson Center 1425.
09:00 – 09:30 | Welcome and Coffee
09:30 – 10:20 | Vern Paulsen
10:30 – 11:00 | Coffee Break
11:00 – 11:50 | Qiang Zeng
12:00 – 02:00 | Lunch
02:00 – 02:50 | Jonathan Novak
03:00 – 03:30 | Coffee Break
03:30 – 04:20 | Benjamin Hayes
04:30 – 05:20 | Michael Hartglass
05:30 – 05:30 | Beer & Pizza
09:00 – 09:30 | Coffee
09:30 – 10:20 | Ken Dykema
10:30 – 11:00 | Coffee Break
11:00 – 11:50 | Brent Nelson
12:00 – 12:50 | Ian Charlesworth
Titles and Abstracts
Ian Charlesworth, University of California, Berkeley
Infinitesimal bi-free probability
Bi-free probability was introduced by Voiculescu in 2013 in order to study the interplay between left and right representations of algebras on their free product space. The cumulants appropriate to this model are derived from the bi-non-crossing partitions, which are twisted copies of the lattice of non-crossing partitions able to keep track of the sidedness of variables. By pulling this twist back to the level of Coxeter groups, I will demonstrate a suitable analogue for Type B bi-non-crossing partitions. The non-crossing partitions of Type B have been used to construct cumulants which describe infinitesimal freeness, i.e., the satisfaction of the vanishing alternating moment condition for freeness to order $t$ under the state $\varphi + \varphi’$. I will show that the same connection holds in the bi-free setting, and also describe an alternating moment condition for infinitesimal bi-free independence. Time permitting I will discuss some further properties of infinitesimal distributions. This project is joint work with several students I have been working with this summer: Zhiheng Li, Kyle P. Meyer, Drew T. Nguyen, Jennifer Pi, and Anna K. Raichev. [Notes]
Ken Dykema, Texas A&M University
Schur-type upper triangular forms and decomposability in finite von Neumann algebras
The spectrum of an operator contains essential information. Even better is when we can find invariant subspaces that break the operator into pieces with conditions on the spectrum. This is the meaning of decomposability of an operator, in the sense of Foias.
An arbitrary element of a finite von Neumann algebra has a sort of spectral distribution measure called its Brown measure. Haagerup and Schultz proved existence of invariant subspaces that break up such an element according to its Brown measure. These Haagerup-Schultz subspaces have been used to provide Schur-type upper triangular forms for such elements. In this talk, we review these constructions and describe how these Schur-type upper triangular forms relate to decomposability in the sense of Foias. [Notes]
Michael Hartglass, Santa Clara University
Non-tracial von Neumann algebras from weighted graphs
Given a weighted graph, I will present a construction of a von Neumann algebra associated to it. This von Neumann algebra will often be non-tracial and will have a summand isomorphic to an almost periodic free Araki-Woods factor. An application to infinite-index subfactors will be discussed. This is joint work with Brent Nelson. [Notes]
Benjamin Hayes, University of Virginia
A Random Matrix approach to the Peterson-Thom conjecture
The Peterson-Thom conjecture states that given an amenable subgalebra of a free group factor, there is a unique maximal amenable subalgebra containing it (uniqueness is the hard part). Equivalently, given two diffuse, amenable subalgebras with diffuse intersection, the algebra they generate is amenable. I will discuss some random matrix conjectures related to the Haagerup-Thorbjornsen theorem that which would imply the Peterson-Thom conjecture. [Notes]
Brent Nelson, Vanderbilt University
Free Stein discrepancy as a regularity condition
Given an n-tuple of non-commutative random variables, its free Stein discrepancy relative to the semicircle law measures how “close” the distribution is to the semicircle law. By considering free Stein discrepancies relative to a broader class of laws, one can define a quantity called the free Stein information. In this talk, we will discuss this and its relation to other free probabilistic quantities such as the free Fisher information and the non-microstates free entropy dimension. This is based on joint work in progress with Ian Charlesworth. [Notes]
Jonathan Novak, University of California, San Diego
Approximating the R-transform
The R-transform is one of the basic special functions of free probability, and as such it behooves us to understand it from several different points of view. I’ll discuss the approximation approach to the R-transform, in which it arises as the large N limit of the Laplace transform of uniform measure on adjoint orbits of $U(N)$. [Slides]
Vern Paulsen, University of Waterloo
Quantum probabilities, synchronous games, and C*-algebras
Many games have higher winning probabilities if the players are allowed to use random answers generated by quantum events than can be achieved by using classical random events. In fact, some games can be won always using such quantum strategies in cases when no such perfect classical strategy exists.
Currently there are several mathematical models that attempt to characterize the set of quantum probability densities and it has only been recently proven that some of these models yield different sets of densities. It is known that whether or not two of these models yield the same set of densities is equivalent to Connes’ embedding conjecture.
In this talk we will review the above ideas and then introduce synchronous games.
Each synchronous game has a *-algebra affiliated with it whose representation theory determines whether or not the game has a perfect strategy belonging to each of these different sets of densities. For the “graph isomorphism game” this algebra is a quotient of the quantum permutation group and results from game theory yield some new information about this quantum group. [Slides]
Qiang Zeng, Queens College, City University of New York
Replica symmetry breaking for mean field spin glass models
In statistical physics, the study of spin glasses was initialized to describe the low temperature state of a class of magnetic alloys in the 1960s. Since then spin glasses have become a paradigm for highly complex disordered systems. Mean field spin glass models were introduced as an approximation of the physical short range models in the 1970s. The typical mean field models include the Sherrington-Kirkpatrick (SK) model, the (Ising) mix p-spin model and the spherical mixed p-spin model.
Starting in 1979, the physicist Giorgio Parisi wrote a series of ground breaking papers introducing the idea of replica symmetry breaking (RSB), which allowed him to predict a solution for the SK model by breaking the symmetry of replicas infinitely many times at low temperature. This is known as full-step replica symmetry breaking (FRSB). In this talk, we will show that Parisi’s FRSB prediction holds at zero temperature for the more general mixed p-spin model. On the other hand, we will show that there exist two-step RSB spherical mixed spin glass models at zero temperature, which are the first examples beyond the replica symmetric, one-step RSB and FRSB phases. Some open problems will be mentioned.
This talk is based on joint works with Antonio Auffinger (Northwestern University) and Wei-Kuo Chen (University of Minnesota). [Slides]