Abstracts
Marie-Charlotte Brandenburg (MPI MiS)
Competitive Equilibrium and Lattice Polytopes
The question of existence of a competitive equilibrium is a game theoretic question in economics. It can be posed as follows: In a given auction, can we make an offer to all bidders, such that no bidder has an incentive to decline our offer?
We consider a mathematical model of this question, in which an auction is modelled as weights on a simple graph. In this model, the existence of an equilibrium can be translated to a condition on certain lattice points in a lattice polytope.
In this talk, we discuss this translation to the polyhedral language. Using polyhedral methods, we show that in the case of the complete graph a competitive equilibrium is indeed guaranteed to exist.
This is joint work with Christian Haase and Ngoc Mai Tran.
Luis Ferroni (KTH)
Lattice points in slices of rectangular prisms
Hypersimplices are ubiquitous within algebraic combinatorics. The problem of calculating its volume, which happens to be an Eulerian number, has motivated much research in the past decades. In this talk we will address the Ehrhart theory of a much more general version of hypersimplices. We will explain how to count the number of lattice points in dilations of certain slices of rectangular prisms. In particular, we will see that these polytopes are polypositroids and are Ehrhart positive. We will also discuss a combinatorial interpretation of the entries of the h*-vector, and we will explain how this can be used to settle the problem of understanding combinatorially the Hilbert series of all algebras of Veronese type. This is joint work with Daniel McGinnis.
Akihiro Higashitani (Osaka)
Lattice polytopes with small numbers of facets arising from combinatorial objects
There are several kinds of lattice polytopes arising from combinatorial objects, e.g., order polytopes, chain polytopes, edge polytopes, matroid polytopes, and so on. In this talk, we introduce some of them and discuss when those families coincide up to unimodular equivalence. In particular, we focus on the case where the number of facets is small, e.g., (d+2), (d+3) or (d+4) facets, where d is the dimension of the polytope. This talk is based on the joint work with Koji Matsushita.
Martina Juhnke-Kubitzke (Osnabrück)
On the gamma-vector of symmetric edge polytopes
Symmetric edge polytopes are a class of lattice polytopes that has seen a surge of interest in recent years for their intrinsic combinatorial and geometric properties as well as for their relations to metric space theory, optimal transport and physics, where they appear in the context of the Kuramoto synchronization model. In this talk, we study 𝛄–vectors associated with h*-vectors of symmetric edge polytopes both from a deterministic and a probabilistic point of view. On the deterministic side, nonnegativity of 𝛄₂ for any graph is proven and the equality case 𝛄₂ = 0 is completely characterized. The latter also confirms a conjecture by Lutz and Nevo in the realm of symmetric edge polytopes. On the probabilistic side, it is shown that the 𝛄–vectors of symmetric edge polytopes of most Erdős–Rényi random graphs are asymptotically almost surely nonnegative up to any fixed entry. This proves that Gal's conjecture holds asymptotically almost surely for arbitrary unimodular triangulations in this setting. This is joint work with Alessio D'Alí, Daniel Köhne and Lorenzo Venturello.
Sophie Rehberg (FU Berlin)
Rational Ehrhart Theory
The Ehrhart quasipolynomial of a rational polytope P encodes fundamental arithmetic data of P, namely, the number of integer lattice points in positive integral dilates of P. Ehrhart quasipolynomials were introduced in the 1960s. They satisfy several fundamental structural results and have applications in many areas of mathematics and beyond. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by Linke (2011), Baldoni-Berline-Koeppe-Vergne (2013), and Stapledon (2017). We introduce a generating-function ansatz for rational Ehrhart quasipolynomials, which unifies several known results in classical and rational Ehrhart theory. In particular, we define y-rational Gorenstein polytopes, which extend the classical notion to the rational setting. This is joint work with Matthias Beck and Sophia Elia.
Francisco Santos (Cantabria)
Empty simplices of large width
The "flatness theorem” states that the maximum lattice width among all hollow convex bodies in ℝd is bounded by a constant Flt(d) depending solely on d. For general K the best current bound is Flt(d) ≤ O(d4/3) (modulo a polylog term) [Rudelson 2000], but for simplices (among other cases) width is known to be bounded by O(d log d) [Banaszczyk et al. 1999]. In contrast, no construction of convex bodies of width more than linear is known.
We show two constructions leading to the first known empty simplices (lattice simplex in which vertices are the only lattice points) of width larger than their dimension:
We introduce cyclotomic simplices and exhaustively compute all the cyclotomic 10-simplices of volume up to 231. Among them we find five empty ones of width 11, and none of larger width.
Using circulant matrices of a specific form, we construct empty d-simplices of width growing asymptotically as d/arcsinh(1) ~ 1.1346 d.
This is joint work with Joseph Doolittle, Lukas Katthän and Benjamin Nill. See arXiv:2103.14925 for details.
Rainer Sinn (Leipzig)
h*-vectors of alcoved lattice polytopes
We discuss unimodality of the h*-vector for alcoved lattice polytopes (of Lie type A). The main ingredient is a fairly explicit triangulation for which we need the assumption that the facets have lattice distance one to the set of interior lattice points. This is joint work with Hannah Sjöberg.
Akiyoshi Tsuchiya (Tokyo )
Ehrhart theory on adjacency polytopes
PQ-type and PV-type adjacency polytopes are lattice polytopes arising from finite graphs. PQ-type adjacency polytopes are isomorphic to root polytopes and their normalized volumes give an upper bound on the number of solutions to algebraic power-flow equations in an electrical network corresponding to their underlying graphs. On the other hand, PV-type adjacency polytopes are also called symmetric edge polytopes and their normalized volumes give an upper bound on the number of solutions to Kuramoto equations, which models the behavior of interacting oscillators. In this talk, we study the h*-polynomials of adjacency polytopes. In particular, for several families of graphs, we give formulas of the h*-polynomials and the normalized volumes of these polytopes in terms of their underlying graphs. This is joint work with Hidefumi Ohsugi.