research interests

My research interests are in asymptotic representation theory and its connections to algebraic combinatorics, random matrix theory and mathematical physics.

research papers

8. Elements of the q-Askey scheme in the algebra of symmetric functions (joint with G. Olshanski), submitted. (arxiv:1808.06179).

7. Universal Behavior of the Corners of Orbital Beta Processes, submitted. (arxiv:1807.06134).

6. Interpolation Macdonald Operators at Infinity, Advances in Applied Mathematics, vol. 101, 2018, 15-59. (arxiv:1712.08014).

5. Pieri Integral Formula and Asymptotics of Jack Unitary Characters, Selecta Mathematica, New Series, vol. 24, no. 3 (2018), 2737-2789. (arxiv:1704.02430).

4. Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q, t)-Gelfand-Tsetlin graph, SIGMA 14 (2018), 001. (arxiv:1704.02429).

3. BC Type Z-measures and Determinantal Point Processes, Advances in Mathematics 334 (2018), 1-80. (arxiv:1701.07060).

2. Markov processes on the duals to infinite-dimensional Lie groups, Annales de l'Institut Henri Poincare: Probabilites et Statistiques, vol. 54, no. 3, 2018. (arxiv:1608.02281).

1. Laurent Phenomenon Sequences (joint with J. H. Alman and J. Huang.) Journal of Algebraic Combinatorics (2013): 1-45. (arxiv:1309.0751)

comments on papers

8. Elements of the q-Askey scheme in the algebra of symmetric functions
This is the first paper of a big project with Prof. Grigori Olshanski. The main object are the quantum BC type z-measures, which are point processes on q-lattices of the real line. They are quantizations of probability measures arising in the problem of harmonic analysis of infinite-dimensional groups. In this context, the measures serve as orthogonality measures for analogues of the q-Racah symmetric functions.

7. Universal Behavior of the Corners of Orbital Beta Processes
There is a relation between the Bessel functions of type A and random matrix theory, that has been known since Neretin's paper: arxiv:math/0301070; see also Gorin-Marcus (arxiv:1706.07393). I wanted to use the theory of Bessel functions to obtain new results about random matrices. Combining the well-known theory of the Dunkl transform, and degenerations of the formulas in my paper 5, I obtained a universality result for orbital beta processes. The orbital beta processes depend on a parameter \beta > 0 and should be thought of as a building block of the well-known beta ensemble.

6. Interpolation Macdonald Operators at Infinity
This is a direct "inhomogeneous" generalization of Nazarov-Sklyanin paper: (arxiv:1212.2960). We give explicit formulas for a hierarchy of operators A^1, A^2, ... that diagonalize the interpolation Macdonald functions. The answer is given in terms of Hall-Littlewood functions and an inhomogeneous analogue of these functions. An unexpected feature is the existence of a vertex operator form for A^1, similar to that in the notes of Negut (arxiv:1310.3515, Thm. 1.2 for n = 1). This suggests there may be a connection between the shuffle algebra and the hierarchy of operators that is in my paper. I think studying this coincidence further would be very interesting.

5. Pieri Integral Formula and Asymptotics of Jack Unitary Characters
After I obtained the formulas in my paper 4, my advisor A. Borodin suggested to find a more direct proof of the asymptotics of Jack polynomials, obtained by Okounkov-Olshanski (arxiv:1301.0634). The difficulty was that the multivariate formula from paper 4 does not degenerate to Jack polynomials. I came up with the "Pieri integral formulas", as a replacement to the multivariate formulas. They have the added advantage that k does not need to be integer (t = q^k). G. Olshanski later shared with me a representation-theoretic interpretation of the Pieri integral formula, and it is section 3.2 of the paper.

4. Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q, t)-Gelfand-Tsetlin graph
We generalize the integral representations and multivariate formulas from the paper of Gorin-Panova (arxiv:1301.0634) to the context of Macdonald polynomials; thus effectively adding two parameters q, t to their theorems. I followed the idea of A. Okounkov, that was printed in Gorin-Panova's article. Though his idea took me as far as the case t = q^k, k integer, a use of Carlson's lemma shows the integral representations extend for any complex k, with Re(k) > 0. Late in 2016, G. Olshanski suggested to study the boundary of the (q, t)-GT graph as an application of my formulas. I characterized said boundary result in the case t = q^k, k integer. This is a generalization of the boundary problem when t = q; that was solved by V. Gorin (arxiv:1011.1769).

3. BC Type Z-measures and Determinantal Point Processes
For other probability measures in asymptotic rep. theory, the most important result is the interpretation of these measures in terms of some point process and the calculation of their correlation kernels. We solve this question for BC type z measures; as a consequence, we solve the problem of harmonic analysis for some infinite symmetric spaces. A remarkable fact is that the answer is given by the same kernel (after formal identification) as in an article of Borodin-Olshanski (http://annals.math.princeton.edu/2005/161-3/p05).

2. Markov processes on the duals to infinite-dimensional Lie groups
In this paper we construct a Markov process presserving the BC type z-measures, which are 4-parameter probability measures arising from the representation theory of the infinite dimensional symplectic and orthogonal groups. The project was suggested by G. Olshanski and A. Borodin in 2015. The processes can be interpreted as stochastic dynamics preserving a measure on point configurations on (0, +\infty) - {1}. I proved also that the measure on point configurations is a determinantal point process (see paper 3). Thus the processes we construct can be seen as a kind of dualization of Brownian motion on Lie groups of type B, C and D.

1. Laurent Phenomenon Sequences (w/ J. H. Alman and J. Huang)
This paper was the result of a summer REU at the University of Minnessota, Twin Cities. We were mentored by Pasha Pylyavskyy. In the project, we made use of Laurent Phenomenon Algebras, introduced by T. Lam and P. Pylyavskyy, to study recurrence relations of certain type that define sequences x_1, x_2, x_3,... with the Laurent phenomenon.