PUBLICATIONS BY SUBJECT
MICHAEL ANSHELEVICH
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.
Noncommutative stochastic processes.
I have investigated numerous types of noncommutative stochastic processes. These include processes with freely independent increments, qLevy 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.

Generators of some noncommutative stochastic processes, Probab. Theory Related Fields 157 (2013), 777815.

Twostate free Brownian motions, J. Funct. Anal. 260 (2011), 541565.

qLévy processes, J. Reine Angew. Math. 576 (2004), 181207.

Itô formula for free stochastic integrals,
J. Funct. Anal. 188 (2002), 292315.

Free stochastic measures via noncrossing partitions II, Pacific J. Math. 207 (2002), 1330.

Partitiondependent stochastic measures and qdeformed cumulants, Doc. Math. 6 (2001), 343384.

Free stochastic measures via noncrossing partitions, Adv. Math. 155 (2000), 154179.
Convolutions and limit theorems.
Distributions of noncommutative stochastic processes form convolution semigroups. I have studied such semigroups in numerous settings, including free and monotone convolutions, and their extensions to the operatorvalued framework. Papers in this section use, in addition to combinatorics, complexanalytic tools.

OperatorValued Monotone Convolution Semigroups and an Extension of the BercoviciPata Bijection (with John D. Williams), arXiv:1412.1413 [math.OA].

Operatorvalued Jacobi parameters and examples of operatorvalued distributions (with John D. Williams), arXiv:1412.1280 [math.OA].

Local limit theorems for multiplicative free convolutions (with JC Wang and Ping Zhong), J. Funct. Anal. 267 (2014), 34693499.

Limit theorems for monotonic convolution and the Chernoff product formula (with
John D. Williams), Int. Math. Res. Notices 2014 (11), 29903021.

Free evolution on algebras with two states II, arXiv:1204.0289 [math.OA].

Convolution powers in the operatorvalued framework (with Serban T. Belinschi, Maxime Fevrier, and Alexandru Nica), Trans. Amer. Math. Soc. 365 (2013), 20632097.

Free evolution on algebras with two states, J. Reine Angew. Math. 638 (2010), 75101.
Combinatorics. Singlevariable 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 noncommuting variables, which get their own section.
Noncommutative polynomial families.
I have investigated orthogonal polynomials in multiple noncommuting variables. I also introduced, in the multivariate noncommutative 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.

Producttype noncommutative polynomial states, Noncommutative Harmonic Analysis with Applications to Probability II, Banach Center Publ., vol. 89, Polish Acad. Sci. Inst. Math., Warsaw, 2010, pp. 4559.

Appell polynomials and their relatives III. Conditionally free theory, Illinois J. Math. 53 (2009), 3966.

Appell polynomials and their relatives II. Boolean theory, Indiana Univ. Math. J. 58 (2009), 929968.

Monic noncommutative orthogonal polynomials, Proc. Amer. Math. Soc. 136 (2008), 23952405.

Orthogonal polynomials with a resolventtype generating function, Trans. Amer. Math. Soc. 360 (2008), 41254143.

Free Meixner states,
Commun. Math. Phys. 276 (2007), 863899.

Appell polynomials and their relatives, Int. Math. Res. Not. 2004 n. 65, 34693531.
Other.
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.

Quantum free YangMills on the plane (with Ambar N. Sengupta), J. Geom. Phys. 62 (2012), 330343.

Free infinite divisibility for qGaussians (with Serban Teodor Belinschi, Marek Bożejko, and Franz Lehner), Math. Res. Lett. 17 (2010), 905916.

Zimmermann type cancellation in the free Faà di Bruno algebra (with Edward G. Effros and Mihai Popa), J. Funct. Anal. 237 (2006), 76104.

The linearization of the central limit operator in free probability theory,
Probab. Theory Related Fields 115 (1999), 401416.

