Ian Charlesworth

Lecturer

Cardiff University Department of Mathematics

Email: charlesworthi@cardiff.ac.uk

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.

## On Free Stein Dimension, with B. Nelson; arXiv:2201.00062 (2022).

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.

## Analogues of Entropy in Bi-Free Probability Theory: Microstates, with P. Skoufranis; IMRN.[arXiv]

In this paper, we extend the notion of microstate free entropy to the bi-free setting. In particular, using the bi-free analogue of random matrices, microstate bi-free entropy is defined. Properties essential to an entropy theory are developed, such as the behaviour of the entropy when transformations on the left variables or on the right variables are performed. In addition, the microstate bi-free entropy is demonstrated to be additive over bi-free collections and is computed for all bi-free central limit distributions.## Free Stein irregularity and dimension, with B. Nelson; J. Operator Theory

**85**(2021), no. 1, 101-133.[arXiv]We introduce a free probabilistic quantity called free Stein irregularity, which is defined in terms of free Stein discrepancies. It turns out that this quantity is related via a simple formula to the Murray--von Neumann dimension of the closure of the domain of the adjoint of the non-commutative Jacobian associated to Voiculescu's free difference quotients. We call this dimension the free Stein dimension, and show that it is a $*$-algebra invariant. We relate these quantities to the free Fisher information, the non-microstates free entropy, and the non-microstates free entropy dimension. In the one-variable case, we show that the free Stein dimension agrees with the free entropy dimension, and in the multivariable case compute it in a number of examples.## Matrix models for ε-free independence, with B. Collins; Arch. Math.

**116**(2021), no. a, 585-605.[arXiv]We investigate tensor products of random matrices, and show that independence of entries leads asymptotically to ε-free independence, a mixture of classical and free independence studied by Młotkowski and by Speicher and Wysoczański. The particular ε arising is prescribed by the tensor product structure chosen, and conversely, we show that with suitable choices an arbitrary ε may be realized in this way. As a result we obtain a new proof that $\mathcal{R}^\omega$-embeddability is preserved under graph products of von Neumann algebras, along with an explicit recipe for constructing matrix models.## Analogues of Entropy in Bi-Free Probability Theory: Non-Microstate, with P. Skoufranis; Adv. Math.

**375**(2020), 107367.[arXiv]In this paper, we extend the notion of non-microstate free entropy to the bi-free setting. Using a diagrammatic approach involving bi-non-crossing diagrams, bi-free difference quotients are constructed as analogues of the free partial derivations. Adjoints of bi-free difference quotients are discussed and used to define bi-free conjugate variables. Notions of bi-free Fisher information and non-microstate entropy are defined and properties of free entropy are extended to the bi-free setting.## Distortion for multifactor bimodules and representations of multifusion categories, with M. Bischoff, S. Evington, L. Giorgetti, and D. Penneys; arXiv:2010.01067 (2020).

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.

## Simultaneous upper triangular forms for commuting operators in a finite von Neumann algebra, with K. Dykema, F. Sukochev, and D. Zanin; Canad. J. Math.

**72**(2019), no. 5, 1188-1245.[arXiv]The joint Brown measure and joint Haagerup--Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are investigated, and several natural properties are proved for these. It is shown that the support of the joint Brown measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved and the joint Brown measure and joint Haagerup--Schultz projections are shown to be have well under the Arens multivariate holomorphic functional calculus of such a commuting tuple.## An alternating moment condition for bi-freeness; Adv. Math.

**346**(2019), 546-568.[arXiv]In this note we demonstrate an equivalent condition for bi-freeness, inspired by the well-known "vanishing of alternating centred moments" condition from free probability. We show that all products satisfying a centred condition on maximal monochromatic \chi-intervals have vanishing moments if and only if the family of pairs of faces they come from is bi-free, and show that similar characterisations hold for the amalgamated and conditional settings. In addition, we construct a bi-free unitary Brownian motion and show that conjugation by this process asymptotically creates bi-freeness; these considerations lead to another characterisation of bi-free independence.## Free entropy dimension and regularity of non-commutative polynomials, with D. Shlyakhtenko; J. Funct. Anal.

**271**(2016), no. 8, 2274-2292.[arXiv]We show that the spectral measure of any non-constant non-commutative polynomial evaluated at a non-commutative \(n\)-tuple cannot have atoms if the free entropy dimension of that \(n\)-tuple is \(n\) (see also work of Mai, Speicher, and Weber). Under stronger assumptions on the \(n\)-tuple, we prove that the spectral measure of any non-constant non-commutative polynomial function is not singular, and measures of intervals surrounding any point may not decay slower than polynomially as a function of the interval's length.## On two-faced families of non-commutative random variables, with B. Nelson and P. Skoufranis; Canad. J. Math.

**67**(2015), no. 6, 1290-1325.[arXiv]We demonstrate that the notions of bi-free independence and combinatorial-bi-free independence of two-faced families are equivalent using a diagrammatic view of bi-non-crossing partitions. These diagrams produce an operator model on a Fock space suitable for representing any two-faced family of non-commutative random variables. Furthermore, using a Kreweras complement on bi-non-crossing partitions we establish the expected formulas for the multiplicative convolution of a bi-free pair of two-faced families.

## Free Stein Dimension. A talk recorded at the

*Wales MPPM seminar*.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.

## Asymptotic ε-independence. A presentation recorded at the Banff International Research Station in October 2019 during the workshop

*Classification Problems in von Neumann Algebras*.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.

## Free Stein Information. A presentation recorded at the Fields Institute in February 2019 during the

*Southern Ontario Operator Algebras Seminar*.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.

## An alternating moment condition and liberation for bi-freeness. A presentation recorded at the Banff International Research Station in December 2016 during the workshop

*Analytic Versus Combinatorial in Free Probability*.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.