Research

Free probability theory is the study of non-commutative random variables. This includes literal random structures whose multiplication is non-commutative, such as random matrices; it also includes objects that are merely interpreted as random variables, such as linear operators over a Hilbert space, where “expected value” is defined by a positive linear functional over the entire algebra of operators.

My personal interests in free probability are three-fold:

• Applications to random matrix theory (and in turn, their applications to data science, particularly in dimension-reduction techniques).
• Applications to operator algebras, particularly namely von Neumann algebras.
• Combinatorial interpretations of free probability (such as free and Boolean cumulants) using lattice paths and non-crossing partitions.

Preprint Articles

(Both papers are currently undergoing journal review.)

Fock representation of free convolution powers

Anshelevich, M.; and Mashburn, J. Available at arXiv:2207.12481

Some Fock spaces with depth two action

Anshelevich, M.; and Mashburn, J. Available at arXiv:2103.13936

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