Here you will find some information about my research interests, our research group and their activities.

Research

Most of my research is on algebraic and enumerative aspects of combinatorics. I am particularly interested in interactions between combinatorics and other fields of mathematics such as representation theory, probability, mathematical physics and enumerative geometry. I like it when combinatorial objects can be used to give an explicit meaning to complicated and abstract structures in mathematics. There are two types of combinatorial objects that I personally have found to be fascinating, as they seem to connect various areas of mathematics. These objects are Young diagrams, and combinatorial maps, which can be interpreted as discrete surfaces.

A large part of my research is supported by the NCN grant project “One-parameter deformations in symmetric functions theory”. The overall goal is to extend our understanding of classical Schur-related discrete models to the realm of their one-parameter deformations. We intend to use combinatorial methods to explain many mysterious phenomenas in a uniform way. Here are some key words that appear in my research:

  • Young diagrams
  • combinatorial maps aka ribbon graphs
  • asymptotic representation theory
  • integrable probability
  • Jack polynomials
  • topological recursion

Research group at IM PAN Kraków

I work in the Kraków branch of IM PAN, where we have a group working on combinatorics and interactions (mathematical physics, probability, representation theory). The group is financially supported by different grants, largely by the NCN grant “One-parameter deformations in symmetric functions theory”.

Postdocs:

Students:

  • Maciej Kowalski (2018-now)
  • Andrey Naradzetski (2023-now)

Activities

Our group has an informal reading seminar. This year (2022/2023) we are planning to discuss some recent results lying at the interface between combinatorics and integrable probability. We usually meet on (every other) Wednesday at 11:00. Here is our schedule (to be updated):

Date Topic Speaker Reading
19.12.2023 On Dubrovin’s Quantum Hopf Hierarchy on the Torus: Profiles of Young Diagrams and Wick-Remainder Decomposition Alex Moll (Reed College)
Abstract:

The Hopf equation $\partial_t v + v \partial_x v = 0$, also known as the inviscid Burgers equation, is the simplest example of an infinite-dimensional non-linear classical integrable Hamiltonian system. For periodic initial conditions, this system supports a classical hierarchy of conserved quantities $H_{\ell}(v) = \tfrac{1}{2\pi} \int_0^{2 \pi} v(x)^{\ell} dx$. In 2016, Dubrovin showed that Schur polynomials emerge organically as eigenfunctions of an integrable quantization of these $H_{\ell}(v)$. In this talk, I will recall Dubrovin’s construction from a combinatorial point of view, then (1) restate Dubrovin’s theorem in terms of profiles of Young diagrams using Proposition 2.7 of Ivanov-Olshanski and (2) highlight a property of these operators which I call a Wick-remainder decomposition. This second property plays a key role in a work in progress with Robert Chang (Reed College), as it is from the Wick-remainder decomposition alone that we can prove general law of large numbers and central limit theorems without need of combinatorial calculation of moments. That being said, we hope that the central role of this decomposition in our proof can shed light on ongoing research in symmetric function theory and algebraic combinatorics using operator methods.
10.01.2024 Enumeration of alternating sign matrices Hans Höngesberg
17.01.2024 Random matrix, maps and the KP hierarchy Victor Nador
24.01.2024 Sections 1-3 Maciej Dołęga Notes of Nikos Zygouras
31.01.2024 Sections 4-6 Hans Höngesberg Notes of Nikos Zygouras
14.02.2024 Sections 7-8 + Section 9 Victor Nador Notes of Nikos Zygouras