The study of regularity in free probability boils down to the question of how much information about a *-algebra can be gleaned from probabilistic properties of its generators. Some of the first results in this theme come from the theory of Voiculescu's free entropy: generators satisfying certain entropic assumptions generate von Neumann algebras which are non-Gamma, or prime, or do not admit Cartan subalgebras. Free Stein dimension -- a quantity I introduced with Nelson -- is a more recent quantity in a similar vein, which is robust under polynomial transformations and not trivial for variables which do not embeddable in R^\omega.

In this talk, I will recall the motivation and definition of free Stein dimension, and spend some time focusing on how (approximate) algebraic relations between generators can be used to provide upper bounds on the Stein dimension; of particular interest are commutation, and good behaviour under conjugation, and I will mention how these results apply in some interesting examples. Time permitting I will discuss how free Stein dimension behaves under ``building block'' operations such as direct sums and tensor products with finite dimensional algebras. This is joint work with Brent Nelson.

Many seemingly ad hoc constructions in algebra become simpler and much more natural through the lens of bicategories. In this talk I'll describe a series of papers with Kate Ponto touching on Euler characteristics, 2 dimensional field theories, and topological Hochschild homology, which never could have been written without thinking bicategorically. Particular focus will be put on iterated traces (relating to 2d field theories) and the structure of topological Hochschild homology.
 

I will discuss recent joint work with Hayes wherein we show that any sofic group G that is initially sub-amenable (a limit of amenable groups in Grigorchuk's space of marked groups) admits two embeddings into the universal sofic group S that are not conjugate by any automorphism of S. Time permitting, I will also characterise precisely when two almost homomorphisms of an amenable group G are conjugate, in terms of certain IRS's associated to the two actions of G. One of the applications of this is to recover the result of Becker-Lubotzky-Thom around permutation stability for amenable groups. The main novelty of our work is the usage of von Neumann algebraic techniques in a crucial way to obtain group theoretic consequences.

The classical Diophantine problem of determining  which integers can be written as a sum of two rational cubes has a long history; it includes works of Sylvester,  Selmer, Satgé, Leiman  and the recent work of Alpöge-Bhargava-Shnidman-Burungale-Skinner.  In this talk, we will  use  Selmer groups of elliptic curves and integral binary cubic forms to study some cases of the rational cube sum problem.  This talk is based on  joint works with D. Majumdar, P. Shingavekar and B. Sury.

[pre-talk at 1:20PM]

In this talk we will discuss how to define Brownian motions on a curved space. We will briefly discuss some definitions on Riemannian manifolds and then focus on a construction on Lie groups. If time permits, we will discuss quasi-invariance with respect to left/right-multiplication and how this is related to the geometry of the group. (p.s. a background in probability is not required)

For example, in the case where \calM = BPGL_n, this gives a recipe to extend fibrations in Brauer-Severi varieties, whereas for \calM = [A^1/Gm] this gives a way to extend families of line bundles with a section. The talk is based on a joint work with Andrea Di Lorenzo.