## 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.

## Matrix models for ε-free independence, with B. Collins; arXiv:1910.04343 (2019).

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.## Free Stein irregularity and dimension, with B. Nelson; arXiv:1902.02379 (2019); to appear in J. Operator Theory.

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.## Analogues of Entropy in Bi-Free Probability Theory: Microstates, with P. Skoufranis; arXiv:1902.03874 (2019).

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.

## 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.## 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.## Combinatorics of bi-freeness with amalgamation, with B. Nelson and P. Skoufranis; Comm. Math. Phys.

**338**(2015), no. 2, 801-847.[arXiv]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.

## Analyzing query optimization process: Portraits of join enumeration algorithms, with A. Nica and M. Panju;

*IEEE 28th International Conference on Data Engineering*(2012).Search spaces generated by query optimizers during the optimization process encapsulate characteristics of the join enumeration algorithms, the cost models, as well as critical decisions made for pruning and choosing the best plan. We demonstrate the*JoinEnumerationViewer*which is a tool designed for visualizing, mining, and comparing plan search spaces generated by different join enumeration algorithms when op- timizing same SQL statement. We have enhanced Sybase SQL Anywhere relational database management system to log, in a very compact format, its search space during an optimization process. Such optimization log can then be analyzed by the*JoinEnumerationViewer*which internally builds the logical and physical plan graphs representing complete and partial plans considered during the optimization process. The optimization logs also contain statistics of the resource consumption during the query optimization such as optimization time breakdown, for example, for join enumeration versus costing physical plans, and memory allocation for different optimization structures. The SQL Anywhere Optimizer implements a highly adapt- able, self-managing, optimization process by having several join enumeration algorithms to choose from, each enhanced with different ordering and pruning techniques. The emphasis of the demonstration will be on comparing and contrasting these join enumeration algorithms. The demonstration scenarios will include optimizing SQL statements under various conditions which will exercise different algorithms, pruning and ordering techniques. The generated search spaces and statistics will then be visualized and compared using the*JoinEnumerationViewer*.

- Highly adaptable query optimizaer search space generation process, with A. Nica; U.S. Patent No. 8,429,151 (2013).