5 Talks, 1 Topic: A Day of Combinatorics

On March 18th, 2026, the 5 Talks in Combinatorics thematic day took place in the Joan Maragall Room at the Faculty of Philology and Communication of the University of Barcelona, in the historic building. The event focused on modern combinatorics and its connections with algebra, geometry, information theory, and discrete mathematics. Five international researchers participated in an intensive one-day programme.

1. Carolina Benedetti Velásquez: opening talk and Lluís Santaló Fellow 2026

The first talk was delivered by Carolina Benedetti Velásquez, Associate Professor at the Department of Mathematics of the Universidad de los Andes (Bogotá, Colombia) and Lluís Santaló Fellow 2026—a visiting position created in memory of Lluís Santaló to host Latin American researchers during research stays in Catalonia.

Benedetti is a Colombian mathematician whose work focuses on combinatorics, algebraic structures, and geometry. She studied at the Universidad Nacional de Colombia, completed her MSc at the Universidad de los Andes, and obtained her PhD in 2013 from York University (Toronto, Canada), with the dissertation Combinatorial Hopf algebras and supercharacters. She has been a visiting professor at Michigan State University and is currently co–Executive Director of Mathematical Circles Colombia, as well as a member of the Comunidad Colombiana de Combinatoria and the Gender and Equity Commission.

Combinatorics of some products of (quantum) Schubert polynomials

In her talk, Benedetti introduced Schubert polynomials, a fundamental basis of the polynomial ring over the integers, deeply connected to algebraic geometry through Schubert varieties. She showed how many of their properties can be expressed through permutation combinatorics and presented combinatorial rules for products of both classical and quantum versions.
The work presented is joint with N. Bergeron, L. Colmenarejo, F. Saliola, and F. Sottile.

2. Amanda Montejano — Discrete Versions of the Brunn–Minkowski Inequality

Next, Amanda Montejano (Universidad Nacional Autónoma de México) offered an overview of the various discrete formulations of the classical Brunn–Minkowski inequality, a cornerstone of convex geometry. She discussed the historical development of the problem and highlighted the use of heavy sets as a tool to derive meaningful inequalities in arbitrary dimensions.

3. Oriol Farràs Ventura — Polynomial Secret Sharing Schemes and Algebraic Matroids

The third talk was given by Oriol Farràs Ventura (Universitat Rovira i Virgili), who explored the connection between matroid theory and secret sharing schemes—cryptographic methods in which a secret can be recovered only from certain authorised subsets of information. As shown by Brickell and Davenport, the access structure of ideal schemes determines a matroid.

Farràs presented polynomial secret sharing schemes and proved that, over sufficiently large fields, the access structures of ideal polynomial schemes are determined by algebraic matroids.

4. Yannic Vargas — Nested pre-Lie operads on combinatorial species

After the lunch break, Yannic Vargas (CUNEF Universidad Madrid) introduced the concept of nested pre-Lie operads, a non-associative generalisation of classical operads in which horizontal associativity is replaced by a pre-Lie law. He showed how this structure arises naturally in graphs, trees, and set partitions, and discussed potential extensions to the braid arrangement.

5. Eleni Tzanaki — On interval hypergraphic polytopes

To close the day, Eleni Tzanaki (University of Crete) presented interval hypergraphic polytopes, a special class of hypergraphic polytopes whose hyperedges are intervals. These polytopes can be interpreted as deformations of the associahedron, and Tzanaki showed that their vertex posets correspond exactly to intervals in the Tamari lattice. She also characterised the interval hypergraphs that yield simple polytopes and described a new family of directed trees called weeping willows.

The event highlighted the diversity of perspectives that coexist in modern combinatorics—from deep algebraic structures to applications in information theory, as well as discrete geometry and polytope theory. It offered a vivid snapshot of a field that continues to expand in scope and connections.