Preprints
Quantum Expanders and Quantifier Reduction for Tracial von Neumann Algebras, with Ilijas Farah and David Jekel.
Completely characterized quantifier elimination for tracial von Neumann algebras (the theory of M admits quantifier elimination if and only if every pair of projections of the same trace are automorphically conjugate).
Established a lack of model completeness for a large class of tracial von Neumann algebras (including all Connes-embeddable II_1 factors). The proof of this in the case of II_1 factors uses an explicit random matrix construction: two tuples are built out of Haar unitaries such that their one-quantifier types match while their two-quantifier types differ due to differences in the relative commutant in an ultrapower.
Uniformly Super McDuff II_1 Factors, with Isaac Goldbring, David Jekel, and Srivatsav Kunnawalkam Elayavalli.
Introduced an ultrafilter-free uniform version of the super McDuff property (the central sequence algebra of a II_1 factor is again a II_1 factor), and showed it is closed under elementary equivalence.
Showed large classes of examples satisfy this uniform super McDuff property.
Open Question: Is there an example of a II_1 factor that is super McDuff, but not uniformly super McDuff?
On the First-Order Free Group Factor Alternative, with Isaac Goldbring.
Established a trichotomy of possibilities for the first-order fundamental group of the free group factors.
Established the dichotomy on the level of 2-quantifier theories (either they all have the same two-quantifier theory, or none of them share a common two-quantifier theory).
An Elementary Proof of the Inequality $\chi \leq \chi^*$ for Conditional Free Entropy, with David Jekel.
Proved that the microstates free entropy is bounded above by the non-microstates free entropy, using elementary methods and relationships between the free and classical entropic quantities.
Generalized this inequality to the case of conditional free entropy (conditioning on a fixed subalgebra).
Recordings of Talks
The Inequality $\chi \leq \chi^*$, via the Classical Entropy of Matrix Approximations, given at the Free Probability workshop for the Thematic Program in Operator Algebras and Applications, Fields Institute, Nov 2023.
Uniformly Super McDuff II_1 Factors, given at the Operator Algebras Seminar, Fields Institute, Sept 2023.
Unpublished Notes
Infinitesimal Bi-Free Probability Theory, with Ian Charlesworth, Amy Li, Kyle Meyer, Drew Nguyen, and Anna Raichev.