5 Talks in Combinatorics

Schubert polynomials are a distinguished linear basis of the ring of polynomials over Z. In algebraic geometry, their product corresponds to intersection of (classes of) Schubert varieties in the cohomology of the flag variety. In combinatorics, a lot of their properties can be expressed using combinatorics of permutations. In this talk we will dive into classical and quantum Schubert polynomials and explore some combinatorial rules for their products.
No knowledge of Schubert polynomials will be assumed.
This is joint work with N. Bergeron, L. Colmenarejo, F. Saliola, F. Sottile.

The Brunn–Minkowski inequality is a fundamental result in convex geometry, with deep connections to various areas of mathematics. In particular, within additive combinatorics, the problem of exploring discrete versions of the Brunn–Minkowski inequality has attracted considerable interest in recent years. Although several formulations have been proposed, there is still no clear consensus on which should be regarded as the “best” discrete analogue of the Brunn–Minkowski theorem. In this talk, we will present a general overview of the problem and its historical development. We will also discuss the study of heavy sets as a tool to derive meaningful inequalities in arbitrary dimensions, which in turn reflect the degree of convexity of the sets involved. This is a joint work with Luis Montejano and Oriol Serra.

Interval hypergraphic polytopes form a special class of hypergraphic polytopes in which every hyperedge of the underlying hypergraph is an interval.  These polytopes can be viewed as deformations of the associahedron, and several of their structural properties are naturally reflected in the combinatorics of the latter. It is well known that the vertices of the associahedron naturally biject to  the Tamari lattice.
We show that this perspective extends to the interval hypergraph setting: the vertex posets of interval hypergraphic polytopes correspond precisely to intervals in the Tamari lattice. Finally, we will  characterize the interval hypergraphs for which the associated hypergraphic polytope is simple. In this case, we describe their vertex posets, which form a new and intriguing class of directed trees that we call weeping willows.

This is joint work with Germain Poullot, Felix Gelinas, Jose Bastidas, Vincent Pilaud and Andrew Sack.

A secret sharing scheme is a method to protect a secret. Given a secret, the scheme consists on generating pieces of information, called shares, in such a way that the secret can be recovered from some subsets of shares, called authorized. The collection of authorized subsets is  the access structure of the scheme. Brickell and Davenport proved that the access structure of optimal secret sharing schemes (called ideal) determines a matroid. Conversely, only entropic matroids admit ideal schemes.
In this work we study polynomial schemes, schemes whose shares are computed by polynomials. We show that for large enough fields, the access structure of ideal polynomial schemes is determined by an algebraic matroid. We extend the connections between ideal linear schemes and linear matroids to the algebraic setting, and we give conditions for an algebraic matroid to be linearly representable over some field extension.
This is a joint work with Amos Beimel and Adriana Moya and was published at TCC 2025.

We introduce the notion of nested pre-Lie operads (NPL for short), a non-associative version of operads where the horizontal associativity is replaced by a pre-Lie law. We show how this structure arises naturally on graphs, trees and set partitions. Finally, we discuss possible extensions of NPL structures to the braid arrangement. Part of this talk is joint work with Dominique Manchon, Hedi Regeiba and Imen Rjaiba.