Ian Charlesworth

S.E.W. Visiting Assistant Professor

UC San Diego Department of Mathematics

Office: AP&M 5880C

Email: ilc@math.ucsd.edu

I have been a visiting assistant professor in the Department of Mathematics at the University of California, San Diego since 2017. 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.' Prior to that, I studied Pure Mathematics and Computer Science at the University of Waterloo in Waterloo, Ontario, Canada.

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.

I will sign messages I send upon request, via GPG; my public key is available here.

### Recent Publications & Preprints

• Joint spectral distributions and invariant subspaces for commuting operators in a finite von Neumann algebra, with K. Dykema, F. Sukochev, and D. Zanin; arXiv:1703.05695 (2017).

Analogues of Brown measure and Haagerup-Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are constructed. These are called joint spectral distribution measures and decomposing projections, respectively, and they satisfy properties similar to their single operator precursors (i.e., Brown measure and Haagerup-Schultz projections). It is shown that the support of the joint spectral distribution 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 natural results are shown to hold for the Arens multivariate holomorphic functional calculus of such a commuting tuple.

• An alternating moment condition for bi-freeness; arXiv:1611.01262 (2016).

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.

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.

• Combinatorics of bi-freeness with amalgamation, with B. Nelson and P. Skoufranis; Comm. Math. Phys. 338 (2015), no. 2, 801-847.

In this paper, we develop the theory of bi-freeness in an amalgamated setting. We construct the operator-valued bi-free cumulant functions, and show that the vanishing of mixed cumulants is necessary and sufficient for bi-free independence with amalgamation. Further, we develop a multiplicative convolution for operator-valued random variables and explore ways to construct bi-free pairs of $$B$$-faces.

• On two-faced families of non-commutative random variables, with B. Nelson and P. Skoufranis; Canad. J. Math. 67 (2015), no. 6, 1290-1325.

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.

### Recorded Presentations

• An alternating moment condition and liberation for bi-freeness (video). 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.