Ian Charlesworth
Lecturer
Cardiff University School of Mathematics
I have been a lecturer with the Department of Mathematics at Cardiff University since 2022. I have been a postdoc with the Department of Mathematics at KU Leuven since 2021. Prior to that I held a postdoctoral position at KU Leuven from 2021-2022, an NSF postdoctoral position at UC Berkeley from 2018-2021, and an S.E. Warschawski Visiting Assistant Professorship at UCSan Diego from 2017-2018. My Ph.D. studies took place from 2012-2017 at the University of California, Los Angeles under the supervision of Professor Dimitri Shlyakhtenko; my thesis was titled 'On bi-free probability and free entropy.' I was an undergraduate at the University of Waterloo in Waterloo, Ontario, Canada studying Pure Mathematics and Computer Science.
My research interests lie mostly in the field of free probability and non-commutative probability theory; the field attempts to apply probabilistic techniques to operator algebras, drawing useful analogues from well-known results in probability. I have also dabbled in the study of subfactors and quantum symmetries. In the distant past I have attacked problems in database query optimization.
We show that graph products of non trivial finite dimensional von Neumann algebras are strongly 1-bounded when the underlying $*$-algebra has vanishing first $L^2$-Betti number. The proof uses a combination of the following two new ideas to obtain lower bounds on the Fuglede Kadison determinant of matrix polynomials in a generating set: a notion called 'algebraic soficity' for $*$-algebras allowing for the existence of Galois bounded microstates with asymptotically constant diagonals; a new probabilistic construction of permutation models for graph independence over the diagonal. Our arguments also reveal a probabilistic proof of soficity for graph products of sofic groups.
We establish several properties of the free Stein dimension, an invariant for finitely generated unital tracial $*$-algebras. We give formulas for its behaviour under direct sums and tensor products with finite dimensional algebras. Among a given set of generators, we show that (approximate) algebraic relations produce (non-approximate) bounds on the free Stein dimension. Particular treatment is given to the case of separable abelian von Neumann algebras, where we show that free Stein dimension is a von Neumann algebra invariant. In addition, we show that under mild assumptions $L^2$-rigidity implies free Stein dimension one. Finally, we use limits superior/inferior to extend the free Stein dimension to a von Neumann algebra invariant---which is substantially more difficult to compute in general---and compute it in several cases of interest.
We call a von Neumann algebra with finite dimensional center a multifactor. We introduce an invariant of bimodules over $\rm II_1$ multifactors that we call modular distortion, and use it to formulate two classification results.
We first classify finite depth finite index connected hyperfinite $\rm II_1$ multifactor inclusions $A\subset B$ in terms of the standard invariant (a unitary planar algebra), together with the restriction to $A$ of the unique Markov trace on $B$. The latter determines the modular distortion of the associated bimodule. Three crucial ingredients are Popa's uniqueness theorem for such inclusions which are also homogeneous, for which the standard invariant is a complete invariant, a generalized version of the Ocneanu Compactness Theorem, and the notion of Morita equivalence for inclusions.
Second, we classify fully faithful representations of unitary multifusion categories into bimodules over hyperfinite $\rm II_1$ multifactors in terms of the modular distortion. Every possible distortion arises from a representation, and we characterize the proper subset of distortions that arise from connected $\rm II_1$ multifactor inclusions.
Regularity questions in free probability ask what can be learned about a tracial von Neumann algebra from probabilistic-flavoured qualities of a set of generators. Broadly speaking there are two approaches — one based in microstates, one in free derivations — which with the failure of Connes Embedding are now known to be distinct. The non-microstates approach is not obstructed by non-embeddable variables, but can be more difficult to work with for other reasons. I will speak on recent work with Brent Nelson, where we introduce a quantity called the free Stein dimension, which measures how readily derivations may be defined on a collection of variables. I will spend some time placing it in the context of other non-microstates quantities, and sketch a proof of the exciting fact that free Stein dimension is a *-algebra invariant.
I will discuss ε-independence, which is an interpolation of classical and free independence originally studied by Młotkowski and later by Speicher and Wysoczanski. To be ε-independent, a family of algebras in particular must satisfy pairwise classical or free independence relations prescribed by a $\{0, 1\}$-matrix ε, as well as more complicated higher order relations. I will discuss how matrix models for this independence may be constructed in a suitably-chosen tensor product of matrix algebras. This is joint work with Benoît Collins.
I will speak on recent joint work with Brent Nelson, where we introduce a free probabilistic regularity quantity we call the free Stein information. The free Stein information measures in a certain sense how close a system of variables is to admitting conjugate variables in the sense of Voiculescu. I will discuss some properties of the free Stein information and how it relates to other common regularity conditions.
Bi-free probability is a generalization of free probability to study pairs of left and right faces in a non-commutative probability space. In this talk, I will demonstrate a characterization of bi-free independence inspired by the "vanishing of alternating centred moments" condition from free probability. I will also show how these ideas can be used to introduce a bi-free unitary Brownian motion and a liberation process which asymptotically creates bi-free independence.