Combinatorics and Interactions

Talks from 2024/25:

May 27

Marino Romero (University of Vienna)

q-Chromatic symmetric functions, Macdonald polynomials, and the Stanley-Stembridge Conjecture

Hikita recently gave an elementary basis expansion for Shareshian-Wachs' q-chromatic symmetric function (or chromatic quasisymmetric function) of a unit interval graph. The coefficients appear as ratios of q-analogues, meaning that when q=1, one gets a positive number (or e-positivity). This proved the Stanley-Stembridge Conjecture, which states that the chromatic symmetric function for (3+1)-free graphs is e-positive.

In recent work, we give an expansion of these q-chromatic symmetric functions in terms of Macdonald polynomials, which now involve parameters q and t. One remarkable consequence is that since q-chromatic symmetric functions have no t variables, we can set t to be anything we want, getting different expansions for q-chromatic symmetric functions. One special case is the elementary basis expansion, from which we can also prove the Stanley-Stembridge Conjecture and re-derive Hikita's formula. The vital element is to utilize Carlsson-Mellit's Aqt algebra from the proof of the Shuffle Conjecture,  along with the action of this algebra on the direct sum of (localized) equivariant K-theories of parabolic flag Hilbert schemes, from Carlsson-Gorsky-Mellit's work. This latter action from geometry is very explicit on symmetric functions, requiring no geometric knowledge to understand.

This talk is based on joint work with S. Griffin, A. Mellit, K. Weigl, and J. J. Wen.

May 27

Vladimir Shlyk

The polytope of integer partitions

The topic of the talk is the structure of the polytope of integer partitions. For every positive integer, this polytope is the convex hull of the multiplicity vectors of its partitions. The facets of the polytope are characterized fairly well, so I will focus on results about its vertices. They concern properties of the vertices that distinguish them from other partitions, connections between vertices, and the intriguing behavior of the vertex number function. I will also present results on the vertex structure of the master corner polyhedron, a key mathematical object in integer linear programming. These are the first to continue Ralph Gomory's study of vertices, and the progress was made after studying the polytope of integer partitions.

May 13

Zbigniew Wojciechowski (TU Dresden)

The smallest quantum symmetric pair gl_1 in sl_2 and Type B diagrammatics

We provide a full representation-theoretic, diagrammatic, and combinatorial description of the module category of finite-dimensional representations of the smallest quantum symmetric pair from Letzter’s classification: quantum \mathfrak{gl}_1 inside quantum \mathfrak{sl}_2. During the talk, we encounter “funny vectors,” Temperley–Lieb combinatorics, and diagrammatic bases of Hom-spaces. This talk is based on the paper arXiv:2406.12132, written with Catharina Stroppel, which forms the first chapter of my PhD thesis.

April 29

Özhan Genç (Jagiellonian University)

Homogeneous Instanton Bundles on Grassmannians

Abstract in PDF file: Click here

April 15

Alimzhan Amanov (IM PAN Krakow)

Highest weight vectors of tensors

We study the highest weight vectors for symmetric and alternating spaces over tensors (or hypermatrices with d indices), whose dimensions are given by generalized Kronecker coefficients. We present a unified explicit construction for corresponding spanning sets of highest weight vectors and characterize linear relations among them. We prove that these highest weight vectors satisfy a natural yet nontrivial duality. As applications, we also give conceptual interpretations to power expansions of Cayley's first hyperdeterminant and its dual exterior Cayley form. 

April 1

Sohail Farhangi (Adam Mickiewicz University)

Undecidability in the Ramsey theory of polynomial equations and Hilbert's 10th problem

A central topic of study in arithmetic Ramsey theory is to determine for a given integral domain R which polynomials p \in R[x_1,\cdots,x_n] are partition regular, i.e., when is it the case that for any finite partition R = \bigcup_{i = 1}^\ell C_i, there exists some 1 \le i_0 \le \ell for which C_{i_0} contains a solution to p(x_1,\cdots,x_n) = 0. We reduce this question to a variant of Hilbert's 10th problem over R, and thereby determine the exact lightface complexity of the set of partition regular polynomials for many integral domains R. We will then discuss the combinatorial implications of this set theoretic result. Time permitting, we will discuss analogous results in the realm of density Ramsey theory. In this setting, we were required to prove a compactness principle and a uniformity principle of independent interest for countable cancellative left amenable semigroups.

March 25

Nitin Chidambaram (University of Edinburgh)

CohFTs from r-th roots and integrability

Witten’s conjecture/ Kontsevich’s theorem, which states that the generating function of psi-class intersection numbers is a tau function for the KdV hierarchy, established a close relationship between enumerative geometry and integrable systems. Subsequently, various other examples of Cohomological Field Theories (CohFTs for short) such as Gromov—Witten theory of P^1, Witten r-spin class and FJRW theory have been shown to produce tau functions for different integrable hierarchies. By working on a certain moduli space of r-th roots considered by Chiodo, one can produce a collection of CohFTs \Theta^{r,s} parametrised by an integer s between 1 and r-1. In this talk, I’ll explain that the descendant potential of all of these (non-semisimple) cohomological field theories are r-KdV tau functions, and (time permitting) explain the relationship to W-algebra representations and topological recursion. 

This talk is based partly on arXiv:2205.15621 with E. Garcia-Failde and A. Giacchetto, and work-in-progress with V. Bouchard, A. Giacchetto and S. Shadrin.

February 18

 Justyna Kosakowska (Nicolaus Copernicus University)

Geometry Meets Combinatorics in Birkhoff-Type Problems

"Birkhoff-type problems" is an informal term referring to problems related to the study of subgroups of abelian groups, as well as to problems that have their origins in this context, such as submodules of modules over discrete valuation rings. In this talk, we will focus on the case of invariant subspaces of nilpotent linear operators.

The main aim of this talk is to demonstrate how various combinatorial tools and invariants (e.g., Littlewood–Richardson tableaux, standard Young tableaux) determine certain properties of invariant subspaces of nilpotent linear operators. We will discuss several partial orders defined on these invariants, including both classical ones and new orders that we have introduced. Furthermore, we will illustrate how these tools and invariants can be applied to explore the geometric and algebraic properties of invariant subspaces of nilpotent linear operators.

February 4

Jerzy Weyman (Jagiellonian University)

From quiver representations to cluster categories

In this talk, I will discuss the basics of quiver representations. Then, I will continue to describe my old results with Harm Derksen on semi-invariants of quivers. Finally, I will show how to construct cluster categories for the ADE quivers together with cluster complexes. No previous knowledge of quiver representations will be required.

January 21

Andrzej Grzesik (Jagiellonian University)

Generalized Turán problem for directed cycles

We study the following question: how many directed cycles of length k can an n-vertex oriented graph have if it does not contain a directed cycle of length l as a subgraph? We establish the order of magnitude of the sought maximum for every k and l, and prove its exact value up to a lower error term in the dense case for l large enough. This is joint work with Justyna Jaworska, Bartłomiej Kielak, Piotr Kuc and Tomasz Ślusarczyk.

January 7

Piotr Sułkowski (University of Warsaw)

Knots, quivers and combinatorics

 I will review the status of the knots-quivers correspondence,

which – as the name indicates – is a relation between quiver

representation theory and knot theory. While it follows from physical

considerations and properties of brane systems in string theory, it also

leads to quite non-trivial statements that connect various areas of

mathematics. To start with, it was shown that this correspondence

enables to express various invariants of knots in terms of invariants of

quivers. The invariants in question are also captured by various

combinatorial models; in particular, for torus knots, these invariants

are encoded in models of lattice paths, such as generalized Duchon or

Schroeder paths. These results can be also determined by the topological

recursion. In this talk I will review some of these developments and

present some open problems in this research area.

December 17

Michał Kotowski (University of Warsaw)

The local and global limit of the continuous-time Mallows process

The Mallows process is a process of random permutations whose marginal at time t is the Mallows distribution with parameter t. It can be thought of as interpolating between the identity permutation and the reverse permutation on n elements. We prove that under an appropriate space and time scaling it possesses a global and a local limit. The global limit admits an explicit description in terms of the permuton limit of the Mallows distribution, analyzed by S. Starr, while the local limit is related to the construction of the infinite Mallows distribution on \mathbb{Z} due to Gnedin and Olshanskii. Joint work with Radosław Adamczak.

December 3

Lorenzo Guerrieri (Jagiellonian University)

Licci ideals of codimension 3 and Schur functors

Let R be a regular local (or graded) ring. An ideal of R is said to be licci if it is in the linkage class of a complete intersection. Licci ideals of codimension 1 or 2 coincide with perfect ideals. In codimension 3 the problem of classifying licci ideals among perfect ideals is wide open. In this talk we show how the licci ideals of codimension 3 can be related to certain Schur functors, and their linkage properties can be deduced by combinatorial methods. The main application provides a constructive algorithm to describe the licci ideals of codimension 3 defining rigid algebras. These ones represent generic models for all licci ideals of codimension 3.

November 19

Borys Kuca (Jagiellonian University)

Recent developments in the polynomial Szemerédi theorem

The polynomial Szemerédi theorem of Bergelson and Leibman asserts the existence of a broad class of polynomial patterns in large subsets of integers. Originally proved using soft methods of ergodic theory, it still awaits a quantitative proof that would give bounds for the size of finite sets avoiding given progressions. I will report on the progress in the last few years towards this question and its generalisations.

November 19

Cesar Cuenca (Ohio State University)

The Symplectic Schur Process

We introduce a new symmetric function that plays the role of skew Schur function for symplectic groups. Many combinatorial identities for this function are developed and employed to construct the Symplectic Schur Process. This new probability ensemble shares many desirable properties with the classical Schur process of Okounkov-Reshetikhin and we will explain some of them in this talk, including the property of being a determinantal point process, the connection to the Berele insertion algorithm, and a new Pearcey-type kernel. This talk is based on joint work with Matteo Mucciconi.

November 5

Piotr Dyszewski (University of Wrocław)

Shuffling planar trees and the Jacobian conjecture

We will discuss a combinatorial approach to the Jacobian conjecture, which states that any Keller map (i.e., any polynomial mapping whose Jacobian determinant is a nonzero constant) has a compositional inverse that is also a polynomial. The Jacobian conjecture may be formulated in terms of a subtree shuffling conjecture of d-Catalan trees, i.e., planar d-ary trees. We use the local limit theory of large random trees to show that the subtree shuffling conjecture holds in a certain asymptotic sense, and thereafter prove an approximate version of the Jacobian conjecture, stating that inverses of Keller maps have small power series coefficients for their high-degree terms.

October 23

Mikołaj Frączyk (Jagiellonian University)

Non-abelian expanders and octonions 

In a recent work Dinur and Meshulam defined the non-abelian Cheeger constant for simplicial complexes. This leads to a corresponding notion of non-abelian expanders, which are families of complexes where this constant is bounded away from 0. Roughly speaking, a family of simplicial complexes {X_i} is a non-abelian expander if every “almost cover” (that is branched cover with small branching locus) of X_i is close to a genuine cover. I will give an example of (as far as we know) the first non-abelian expander family of bounded degree simplicial complexes. We will use certain nice triangulations of compact octonionic hyperbolic manifolds. Based on a joint work with Ben Lowe.