My research is in free probability, focused in the non-tracial case (i.e. type III von Neumann algebras), and most of my results are concerning the idea of free transport due to Guionnet and Shlyakhtenko. I also have work related to subfactor planar algebras and the emerging field of bi-free probability.

# Publications

1. Free monotone transport without a trace; Comm. Math. Phys. 334 (2015), no. 3, 1245-1298.
2. Free transport for finite depth subfactor planar algebras; J. Funct. Anal. 268 (2015), no. 9, 2586–2620.
3. Combinatorics of bi-freeness with amalgamation, with I. Charlesworth and P. Skoufranis; Comm. Math. Phys. 338 (2015), no. 2, 801–847.
4. On two-faced families of non-commutative random variables, with I. Charlesworth and P. Skoufranis; Canad. J. Math. 67 (2016), no. 6, 1290-1325.
5. An application of free transport to mixed $q$-Gaussian algebras, with Q. Zeng; Proc. Amer. Math. Soc., 144 (2016), no. 10, 4357-4366.
6. Free montone transport for infinite variables, with Q. Zeng; Int. Math. Res. Not. IMRN, to appear (2017).
7. On finite free Fisher information for eigenvectors of a modular operator; J. Funct. Anal. 273 (2017), no. 7, 2292-2352
8. Free Stein kernels and an improvement of the free logarithmic Sobolev inequality, with M. Fathi; Adv. Math., 317 (2017), no. 7, 193-223.
9. Free transport for interpolated free group factors, with M. Hartglass; J. Funct. Anal. 274 (2017), no. 1, 222-251

# Dissertation

Here is a copy of my dissertation: Non-tracial free transport and applications

# Presentations

A random matrix approach to absorption in free products A video recorded by Fields Institute for Research in Mathematical Sciences in May 2020 during The 48th Canadian Operator Symposium, which occurred virtually.

Free products of finite-dimensional von Neumann algebras and free Araki–Woods factors A video recorded at the Fields Institute for Research in Mathematical Sciences in June 2019 during the workshop on Applications to Random Matrices and Free Probability of Free Noncommutative Functions.

Free Stein kernels and an improvement of the free logarithmic Sobolev inequality (Video) A video recorded at the Banff International Research Station in December 2016 during the workshop Analytic Versus Combinatorial in Free Probability.

On finite free Fisher information for eigenvectors of the modular operator (Video) A video recorded at the Hausdorff Research Institute for Mathematics in May 2016 during the Noncommutative Geometry and Operator Algebras Spring Institute.

Free monotone transport without a trace (Presentation) A presentation of my paper listed above.

Shannon's monotonicity problem for free and classical entropy A presentation from the Optimal Transport and Applications 2013 Summer School of the proof of the monotonicity of free entropy in the identically titled paper by Dimitri Shlyakhtenko and Hanne Schultz. Contains a crash course on free probability and a few simple examples. For a more comprehensive introduction to free probability, consult these notes.

Indecomposability of free nonsingular actions by nonamenable groups in QH_reg A presentation of the proof of Theorem D in Type III factors with unique cartan decomposition by Cyril Houdayer and Stefaan Vaes.

# Notes

Amenable Groups Some notes about amenable groups and equivalent conditions. Adapted from Brown and Ozawa.

Tomita-Takesaki Theory On going notes on Tomita-Takesaki theory, modular automorphism groups, and type III von Neumann algebras. Adapted from Theory of Operator Algebras II, by Masamichi Takesaki.