Publications by subject



I would rather not list future projects on a publicly accessible web page. Come talk to me if you are interested, and I will be very happy to discuss them with you in person.

Instead, here are brief descriptions of my completed work. You may also want to look at some of the talk slides linked below, as well as those on the undergraduate page. If the links don't work for you, search for the same titles on the Mathematics arXiv, where you can also read the abstracts for the articles.

The division by subject is approximate, many papers could easily fit into two or more categories. The connections between different topics mentioned below are probably what I find most interesting about free probability. On the other hand, it goes without saying that to begin research in free probability, knowledge of only one field (functional analysis or probability or combinatorics) is necessary sufficient.

Non-commutative stochastic processes.

I have investigated numerous types of non-commutative stochastic processes. These include processes with freely independent increments, q-Levy processes (which I introduced), and stochastic processes in algebras with two states. Common themes for all these processes include Fock space representations, stochastic integrals, and multivariate polynomials, which are usually three equivalent ways of looking at the same object.

Convolutions and limit theorems.

Distributions of non-commutative stochastic processes form convolution semigroups. I have studied such semigroups in numerous settings, including free and monotone convolutions, and their extensions to the operator-valued framework. Papers in this section use, in addition to combinatorics, complex-analytic tools.

Combinatorics. Single-variable orthogonal polynomials.

I am interested in connections between combinatorics, especially moments, generating functions, Jacobi parameters, and orthogonal polynomials, on one hand, and probability theory, operators, and stochastic processes. This applies especially to the Meixner class of measures, and its analog in free probability, the free Meixner class, which I introduced. I am also interested in polynomials in multiple non-commuting variables, which get their own section.

Non-commutative polynomial families.

I have investigated orthogonal polynomials in multiple non-commuting variables. I also introduced, in the multivariate non-commutative context, the analogs of Appell polynomial families. A related class of objects I am interested in are free Meixner states, the multivariate versions of free Meixner measures.


The articles in this section do not fit precisely into the other sections, although most of them still involve combinatorics. Their subjects include geometry and Hopf algebras.