If $n$ “runners” move along a circle of length one, all starting at the origin, each with its own (constant) velocity, then there is a time at which they are all at distance at least $1/(n+1)$ from the origin.

In its shifted version (sLRC) the runners are allowed to start each at a different position and the velocities are assumed distinct (since, otherwise, giving all runners the same velocity produces an easy counter-example). The conjecture has been approached from different perspectives, and is proved until $n=6$ in the original version (Barajas and Serra 2008) and $n=3$ in the shifted version (Cslovjecsek et al. 2022). The latter is based in a reformulation of LRC and sLRC as questions about the covering radii of certain zonotopes, a connection developed by Malikiosis and Schymura (2017).

In this talk I will review the conjecture and its relation to zonotopes and covering radii, and will show that, both in the original and the shifted versions, if the conjecture holds for integer velocities adding up to at most $n^{2n}$ then it holds for arbitrary velocities. This improves a recent result of Terence Tao, who proved the same with a bound of $n^{Cn^2}$ instead. We then use this bound to computationally prove the shifted version sLRC of the conjecture in the first open case, $n=4$. The first part is joint with Malikiosis and Shymura, and the second part with Alcántara and Criado.

The h-vector of a matroid is an important invariant that has been the subject of intense study in recent years. A still open conjecture of Stanley posits that the h-vector of a matroid is a pure O-sequence, meaning that it can be obtained by counting faces of a pure multicomplex. Merino established Stanley’s conjecture for the case of cographic matroids via chip-firing on graphs and the notion of a G-parking function. Inspired by these constructions, we introduce the notion of a “cycle system” for a matroid M — a family of circuits of M with overlap properties that mimic cut-sets in a graph. A choice of cycle system on M defines a collection of integer sequences that we call “coparking functions”, which we show are in bijection with the set of bases of M. This leads to a proof of Stanley’s conjecture for the case of matroids that admit cycle systems, which, for instance, include graphic matroids of cones as well as K_{3,3}-minor free graphs. Joint work with Scott Corry, Solis McClain, David Perkinson, and Lixing Yi.

A polytope is called indecomposable if it cannot be expressed (non-trivially) as a Minkowski sum of other polytopes. It is not easy to certify the indecomposability of a polytope, and several criteria with increasing strength have been developed since the concept was introduced by Gale in 1954. I’ll present a new approach for proving polytope indecomposability via the (extended) graph of edge dependencies that encompasses most of the previous techniques. Our first application is to provide new family of indecomposable deformed permutahedra that are not matroid polytopes. The problem of characterizing the extreme rays of the submodular cone, that is, indecomposable deformed permutahedra, goes back to Edmonds in 1970, and is wide open. This is joint work with Germain Poullot.

Directional convexity is a refined notion of convexity in which convexity is required only along specific directions. This concept arises naturally in the theory of multivariate calculus of variations and has intriguing applications in cryptography. However, constructing directional convex hulls remains a highly challenging problem. In the few cases where explicit constructions are known, the resulting hulls exhibit rich combinatorial structures, sometimes polyhedral, other times semialgebraic. In this talk, I will review key results and algorithms related to directional convexity and present recent developments from a joint work with Bogdan Raita.

In this talk we will explore the fundamental operation of decomposing a polytope into a Minkowski sum of two other polytopes. This idea leads us to study the set of all possible Minkowski summands, which forms a polyhedral cone. We examine computational methods for constructing these cones, techniques for understanding their facial structure, and applications to combinatorially defined polytopes. This talk will be a survey of the field while highlighting open questions for future research.

Motivated by a recent conjecture that the chain polynomial of any geometric lattice has only real zeros, we study the family of bond lattices of complete bipartite graphs $ K_{m,n}$.

The bond lattice $ \mathcal L_G$ of a graph $G$ is a geometric lattice whose elements are vertex partitions of $G$ in which each block induces a connected subgraph.

The chain polynomial of $ \mathcal L_G$  coincides with  the $f$-polynomial of the order complex $\Delta(\mathcal L_G)$ of $\mathcal L_G$.

Since real-rootedness of the $h$-polynomial implies that of the $f$-polynomial, we focus on the former, for which a wider range of techniques is available.

In particular, we exploit the fact that every geometric lattice  admits an $R$-labelling which, in turn, provides  a combinatorial interpretation of the $h$-polynomial as the distribution of descents  over all maximal chains.

We apply the above framework to complete bipartite graphs $ K_{2,n} $, deriving a surprisingly simple formula for the $ h $-polynomial,  involving Eulerian and bi-colored Eulerian polynomials.

We then discuss how these results extend to the case of  $ K_{3,n} $ with the aim of gaining insight into possible generalizations to the family of all  complete bipartite graphs $K_{m,n}$.

This is joint work in progress with Katerina Kalampogia.

A. Genevois (2022) introduced and investigated mediangle graphs as a common generalization of median graphs (1-sekeleta of CAT(0) cube complexes) and Coxeter graphs (Cayley graphs of Coxeter groups) and studied groups acting on them. He asked if mediangle graphs can be endowed with the structure of a contractible cell complex. In a joint work with K. Knauer we answer this in the affirmative by proving that (bipartite) mediangle graphs are tope graphs of finitary Complexes of Oriented Matroids (COMs). We also show that the oriented matroids (OMs) constituting the cells of COMs arising from mediangle graphs are exactly the simplicial OMs.

In the talk, I will try to explain this result and outline its proof.

The basis graph of a matroid is the graph whose vertices are the bases of the matroid and whose edges are the pairs {A,B} of bases differing by an elementary exchange (i.e., the symmetric difference of A and B is of size 2). Basis graphs of matroids have been characterized by Maurer (1973) as graphs satisfying some local conditions and global metric conditions. In this talk, we will present the result of Maurer and we will explain how we can replace the global metric condition by a topological condition.

(based on joint work with Victor Chepoi et Damian Osajda)

The configuration of a matroid M is the abstract lattice of cyclic flats (flats that are unions of circuits) where we record the size and rank of each cyclic flat, but not the set. From the configuration of M, one can compute well-known matroid invariants such as the Tutte polynomial and the  stronger G-invariant.  We show how, given a matroid M that  satisfies a condition involving non-modular pairs of cyclic flats,  to produce a matroid M’ that is not isomorphic to M but has the same configuration as M.  We then look at this construction within lattice path matroids and show that it can be applied to all LPMs except to those that are fundamental transversal matroids. Hence, most LPMs have a Tutte-equivalent mate (which is not a LPM, actually). We will also introduce the notion of configuration-unique matroid (i.e., a matroid uniquely determined, up to isomorphism, by its configuration) and comment some results on configuration-unique matroids.

Matroid base polytopes are the convex hulls of the indicator vectors of the bases of a matroid. Despite considerable interest, very little was known historically about triangulations of matroid polytopes. I will describe the first nice triangulation of an arbitrary matroid polytope. More precisely, I will describe the first regular unimodular triangulation of a matroid polytope, and I will explain how to extend this triangulation to all integral generalized permutahedra. This is joint work with Gaku Liu.

The famous Tamari lattice is just one of many lattices, whose Hasse diagram is the graph of associahedron. Other such lattices include accordion lattices for different reference triangulations. We introduce an orientation on the edge graph of Kapranov’s permutoassociahedron, so that the resulting poset had accordion lattices of all reference triangulations as subposets (conjecturally, the big poset is a lattice itself). We then introduce two seemingly different yet isomorphic orientations for permutopermutahedron, and thus arrive at the notion of associoassociahedron.

Pointed pseudotriangulations (ppts) of a point configuration in general position have been studied for many years. There exists a polytope (the ppt polytope) whose vertices correspond to the ppts on a fixed general point configuration. When the point configuration forms a convex polygon, the ppt polytope is a realisation of the associahedron. An analogue of the ppt polytope is not yet known for arbitrary point configurations.

In recent work, we interpret ppts in terms of decompositions of objects in a triangulated category into simpler pieces. This point of view offers two advantages.
(1) We naturally expand the notion of ppts to arbitrary point configurations, and we call these “wiggly ppts”.
(2) We understand the behaviour of the associated simplicial complex of ppts under deforming the point set.

Furthermore, motivated by this interpretation, we construct an analogue of the ppt polytope (called the wigglyhedron) for point configurations in which all the points are collinear. This talk is based on joint work with Anand Deopurkar, Anthony Licata, and Vincent Pilaud.