BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Akiyoshi Tsuchiya (Tokyo) DTSTART:20220303T080000Z DTEND:20220303T085000Z DTSTAMP:20260422T212706Z UID:MiniSymLattPoly22/1 DESCRIPTION:Title: Ehrhart theory on adjacency polytopes\nby Akiyoshi Tsuchiya (Tokyo) as part of Mini-Symposium on Lattice Polytopes\n\n\nAbstract\nPQ-t ype and PV-type adjacency polytopes are lattice polytopes arising from fin ite graphs. PQ-type adjacency polytopes are isomorphic to root polytopes a nd 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 polytope s are also called symmetric edge polytopes and their normalized volumes gi ve an upper bound on the number of solutions to Kuramoto equations\, which models the behavior of interacting oscillators. In this talk\, we study t he h*-polynomials of adjacency polytopes. In particular\, for several fami lies of graphs\, we give formulas of the h*-polynomials and the normalized volumes of these polytopes in terms of their underlying graphs. This is j oint work with Hidefumi Ohsugi.\n LOCATION:https://researchseminars.org/talk/MiniSymLattPoly22/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Martina Juhnke-Kubitzke (Osnabrück) DTSTART:20220303T104000Z DTEND:20220303T113000Z DTSTAMP:20260422T212706Z UID:MiniSymLattPoly22/2 DESCRIPTION:Title: On the gamma-vector of symmetric edge polytopes\nby Martina Juhnke-Kubitzke (Osnabrück) as part of Mini-Symposium on Lattice Polytope s\n\n\nAbstract\nSymmetric edge polytopes are a class of lattice polytopes that has seen a surge of interest in recent years for their intrinsic com binatorial and geometric properties as well as for their relations to met ric space theory\, optimal transport and physics\, where they appear in th e context of the Kuramoto synchronization model. In this talk\, we study $ \\gamma$–vectors associated with $h^*$-vectors of symmetric edge polytop es both from a deterministic and a probabilistic point of view. On the det erministic side\, nonnegativity of $\\gamma_2$ for any graph is proven and the equality case $\\gamma_2=0$ is completely characterized. The latter a lso confirms a conjecture by Lutz and Nevo in the realm of symmetric edge polytopes. On the probabilistic side\, it is shown that the $\\gamma$–ve ctors of symmetric edge polytopes of most Erdős–Rényi random graphs ar e asymptotically almost surely nonnegative up to any fixed entry. This pro ves that Gal's conjecture holds asymptotically almost surely for arbitrary unimodular triangulations in this setting. This is joint work with Alessi o D'Alí\, Daniel Köhne and Lorenzo Venturello.\n LOCATION:https://researchseminars.org/talk/MiniSymLattPoly22/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Luis Ferroni (KTH) DTSTART:20220303T130000Z DTEND:20220303T135000Z DTSTAMP:20260422T212706Z UID:MiniSymLattPoly22/3 DESCRIPTION:Title: Lattice points in slices of rectangular prisms\nby Luis Ferr oni (KTH) as part of Mini-Symposium on Lattice Polytopes\n\n\nAbstract\nHy persimplices are ubiquitous within algebraic combinatorics. The problem of calculating its volume\, which happens to be an Eulerian number\, has mot ivated 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 polyt opes are polypositroids and are Ehrhart positive. We will also discuss a c ombinatorial interpretation of the entries of the $h^*$-vector\, and we wi ll explain how this can be used to settle the problem of understanding com binatorially the Hilbert series of all algebras of Veronese type. This is joint work with Daniel McGinnis.\n LOCATION:https://researchseminars.org/talk/MiniSymLattPoly22/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Sophie Rehberg (FU Berlin) DTSTART:20220303T143000Z DTEND:20220303T152000Z DTSTAMP:20260422T212706Z UID:MiniSymLattPoly22/4 DESCRIPTION:Title: Rational Ehrhart Theory\nby Sophie Rehberg (FU Berlin) as pa rt of Mini-Symposium on Lattice Polytopes\n\n\nAbstract\nThe Ehrhart quasi polynomial of a rational polytope $P$ encodes fundamental arithmetic data of $P$\, namely\, the number of integer lattice points in positive integra l dilates of $P$. Ehrhart quasipolynomials were introduced in the 1960s. T hey satisfy several fundamental structural results and have applications i n 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 defi ne y-rational Gorenstein polytopes\, which extend the classical notion to the rational setting. This is joint work with Matthias Beck and Sophia Eli a.\n LOCATION:https://researchseminars.org/talk/MiniSymLattPoly22/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Akihiro Higashitani (Osaka) DTSTART:20220304T080000Z DTEND:20220304T085000Z DTSTAMP:20260422T212706Z UID:MiniSymLattPoly22/5 DESCRIPTION:Title: Lattice polytopes with small numbers of facets arising from comb inatorial objects\nby Akihiro Higashitani (Osaka) as part of Mini-Symp osium on Lattice Polytopes\n\n\nAbstract\nThere are several kinds of latti ce 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 j oint work with Koji Matsushita.\n LOCATION:https://researchseminars.org/talk/MiniSymLattPoly22/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Rainer Sinn (Leipzig) DTSTART:20220304T104000Z DTEND:20220304T113000Z DTSTAMP:20260422T212706Z UID:MiniSymLattPoly22/6 DESCRIPTION:Title: $h^*$-vectors of alcoved lattice polytopes\nby Rainer Sinn ( Leipzig) as part of Mini-Symposium on Lattice Polytopes\n\n\nAbstract\nWe discuss unimodality of the $h^*$-vector for alcoved lattice polytopes (of Lie type $\\mathsf{A}$). The main ingredient is a fairly explicit triangul ation for which we need the assumption that the facets have lattice distan ce one to the set of interior lattice points. This is joint work with Hann ah Sjöberg.\n LOCATION:https://researchseminars.org/talk/MiniSymLattPoly22/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Marie-Charlotte Brandenburg (MPI MiS) DTSTART:20220304T130000Z DTEND:20220304T135000Z DTSTAMP:20260422T212706Z UID:MiniSymLattPoly22/7 DESCRIPTION:Title: Competitive Equilibrium and Lattice Polytopes\nby Marie-Char lotte Brandenburg (MPI MiS) as part of Mini-Symposium on Lattice Polytopes \n\n\nAbstract\nThe question of existence of a competitive equilibrium is a game theoretic question in economics. It can be posed as follows: In a g iven auction\, can we make an offer to all bidders\, such that no bidder h as an incentive to decline our offer?\n\nWe 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 t o a condition on certain lattice points in a lattice polytope.\n\nIn this talk\, we discuss this translation to the polyhedral language. Using polyh edral methods\, we show that in the case of the complete graph a competiti ve equilibrium is indeed guaranteed to exist.\nThis is joint work with Chr istian Haase and Ngoc Mai Tran.\n LOCATION:https://researchseminars.org/talk/MiniSymLattPoly22/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Francisco Santos (Cantabria) DTSTART:20220304T143000Z DTEND:20220304T152000Z DTSTAMP:20260422T212706Z UID:MiniSymLattPoly22/8 DESCRIPTION:Title: Empty simplices of large width\nby Francisco Santos (Cantabr ia) as part of Mini-Symposium on Lattice Polytopes\n\n\nAbstract\nThe "fla tness theorem” states that the maximum lattice width among all hollow co nvex bodies in $\\mathbb{R}^d$ is bounded by a constant $\\operatorname{Fl t}(d)$ depending solely on $d$. For general $K$ the best current bound is $Flt(d) \\le O(d^{4/3})$ (modulo a polylog term) [Rudelson 2000]\, but fo r 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 bodi es of width more than linear is known.\n\nWe show two constructions leadin g to the first known $\\text{\\it empty simplices}$ (lattice simplex in wh ich vertices are the only lattice points) of width larger than their dimen sion:\n\n• We introduce $\\text{\\it cyclotomic simplices}$ and exhausti vely compute all the cyclotomic $10$-simplices of volume up to $2^{31}$.\n Among them we find five empty ones of width $11$\, and none of larger widt h.\n\n• Using $\\text{\\it circulant}$ matrices of a specific form\, we construct empty $d$-simplices of width growing asymptotically as $d/\\ope ratorname{arcsinh}(1) \\sim 1.1346\\\,d$.\n\nThis is joint work with Josep h Doolittle\, Lukas Katthän and Benjamin Nill. See arXiv:2103.14925 for d etails.\n LOCATION:https://researchseminars.org/talk/MiniSymLattPoly22/8/ END:VEVENT END:VCALENDAR