Given a sequence of positive integers s = (s₁, ..., sₙ), the s-lecture hall simplex is defined as Pₙ^s = conv{(0, ..., 0), (0, ..., sₙ), ..., (s₁, ..., sₙ)}. In recent years, several works have studied conditions on s under which the associated Ehrhart polynomial is positive or not (see Question 6.8 in Olsen, 2019).
The objective of this talk is to present a new and surprising family of sequences s = (s₁, ..., sₙ) for which Pₙ^s is not Ehrhart positive. Specifically, we consider s = (a, ..., a, a + 1) ∈ ℤⁿ with a ∈ ℕ and n > 4, and we prove that Pₙ^s is not Ehrhart positive when a is large enough. This is joint work with Martina Juhnke and Germain Poullot.
We study a construction of algebras with duality which model Chow rings and K-rings in algebraic geometry, and surprisingly exhibit many of the properties we would expect coming from geometry. We construct two algebras and relate them via an analogue of the Chern isomorphism, Hirzebruch-Riemann-Roch theorem and study the existence of an exceptional isomorphism. This is joint work with Andreas Gross and Leonid Monin.
A realisation of a graph in the plane as a bar-joint framework is rigid if there are finitely many other realisations, up to isometries, with the same edge lengths. Each of these finitely-many realisations can be seen as a solution to a system of quadratic equations prescribing the distances between pairs of points. For generic realisations, the size of the solution set depends only on the underlying graph so long as we allow for complex solutions. We provide a characterisation of the realisation number– that is the cardinality of this complex solution set– of a minimally rigid graph. Our characterisation uses tropical geometry to express the realisation number as an intersection of Bergman fans of the graphic matroid. In particular, we can make the computations inside the Chow cohomology ring of the permutahedral variety. As a consequence, we derive a combinatorial upper bound on the realisation number involving the Tutte polynomial.
We show that the maximal shifts in the minimal free resolution of a monomial ideal are subadditive as a function of the homological degree, answering a question that has received some attention in the literature in recent years. To do so, we introduce an Eilenberg-Zilber type shuffle product on the simplicial chain complex of lattices, and apply this to study the homological structure of LCM lattices of monomial ideals, resulting in a generalization of the original question. This is joint work with Karim Adiprasito, Anders Björner, Minas Margaritis and Volkmar Welker (https://arxiv.org/abs/2404.16643).
Lattice polytopes are called IDP polytopes if any lattice point in a k-th dilation is a sum of k lattice points in the polytope. It is a long-standing conjecture whether the numerator of the Ehrhart series of an IDP polytope, the so-called h^*-polynomial, has a unimodal coefficient vector. In this minitalk, I will present examples showing that h^*-vectors of IDP polytopes do not have to be log-concave. This is joint work in progress with Vadym Kurylenko and Benjamin Nill.
A result of N.C. Leung and V. Reiner states that certain convexity conditions on a complete rational simplicial fan determine the sign of the signature of the Poincaré pairing on the cohomology of the associated toric variety. In joint work with P. Bressler, we provide a purely "fan-theoretic" proof of their result.
A necessary condition for the sum of complex line bundles to split off of a complex vector bundle is the vanishing of the appropriate virtual Chern classes. In a joint work with Baylee Schutte (arXiv:2411.14161) we prove that this condition is also sufficient for the existence of such splittings for one, two, or three complex line bundles. Our methods are based on Moore-Postnikov theory and elementary rational homotopy theory.
Many enumeration problems in algebra, geometry and number theory can be encoded by a generating series - namely, Dirichlet series - that exhibit properties akin to those of the Riemann zeta function. A prototypical example is the Dedekind zeta function, which enumerates ideals of finite index within the ring of integers of a number field. In this short talk, I will present some examples in which such generating series are given by finitary formulae, expressible in terms of two types of data: the number of rational points on algebraic varieties defined over finite fields, and the enumeration of integral/lattice points on rational polyhedral cones or fans.