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:20260404T094120Z UID:MiniSymLattPoly22/1 DESCRIPTION:Title: Ehrhart theory on adjacency polytopes\nby Akiyoshi Ts uchiya (Tokyo) as part of Mini-Symposium on Lattice Polytopes\n\n\nAbstrac t\nPQ-type and PV-type adjacency polytopes are lattice polytopes arising f rom finite graphs. PQ-type adjacency polytopes are isomorphic to root poly topes and their normalized volumes give an upper bound on the number of so lutions to algebraic power-flow equations in an electrical network corresp onding to their underlying graphs. On the other hand\, PV-type adjacency p olytopes are also called symmetric edge polytopes and their normalized vol umes 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 sever al families of graphs\, we give formulas of the h*-polynomials and the nor malized volumes of these polytopes in terms of their underlying graphs. Th is is joint work with Hidefumi Ohsugi.\n LOCATION:https://stable.researchseminars.org/talk/MiniSymLattPoly22/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Martina Juhnke-Kubitzke (Osnabrück) DTSTART:20220303T104000Z DTEND:20220303T113000Z DTSTAMP:20260404T094120Z UID:MiniSymLattPoly22/2 DESCRIPTION:Title: On the gamma-vector of symmetric edge polytopes\nby M artina Juhnke-Kubitzke (Osnabrück) as part of Mini-Symposium on Lattice P olytopes\n\n\nAbstract\nSymmetric edge polytopes are a class of lattice po lytopes that has seen a surge of interest in recent years for their intrin sic combinatorial and geometric properties as well as for their relations to metric space theory\, optimal transport and physics\, where they appea r in the context of the Kuramoto synchronization model. In this talk\, we study $\\gamma$–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 $\\gamma_2$ for any graph is pro ven and the equality case $\\gamma_2=0$ is completely characterized. The l atter also confirms a conjecture by Lutz and Nevo in the realm of symmetri c edge polytopes. On the probabilistic side\, it is shown that the $\\gamm a$–vectors of symmetric edge polytopes of most Erdős–Rényi random gr aphs are asymptotically almost surely nonnegative up to any fixed entry. T his proves that Gal's conjecture holds asymptotically almost surely for ar bitrary unimodular triangulations in this setting. This is joint work with Alessio D'Alí\, Daniel Köhne and Lorenzo Venturello.\n LOCATION:https://stable.researchseminars.org/talk/MiniSymLattPoly22/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Luis Ferroni (KTH) DTSTART:20220303T130000Z DTEND:20220303T135000Z DTSTAMP:20260404T094120Z UID:MiniSymLattPoly22/3 DESCRIPTION:Title: Lattice points in slices of rectangular prisms\nby Lu is Ferroni (KTH) as part of Mini-Symposium on Lattice Polytopes\n\n\nAbstr act\nHypersimplices are ubiquitous within algebraic combinatorics. The pro blem of calculating its volume\, which happens to be an Eulerian number\, has motivated much research in the past decades. In this talk we will addr ess the Ehrhart theory of a much more general version of hypersimplices. W e will explain how to count the number of lattice points in dilations of c ertain slices of rectangular prisms. In particular\, we will see that thes e polytopes are polypositroids and are Ehrhart positive. We will also disc uss a combinatorial interpretation of the entries of the $h^*$-vector\, an d we will explain how this can be used to settle the problem of understand ing combinatorially the Hilbert series of all algebras of Veronese type. T his is joint work with Daniel McGinnis.\n LOCATION:https://stable.researchseminars.org/talk/MiniSymLattPoly22/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Sophie Rehberg (FU Berlin) DTSTART:20220303T143000Z DTEND:20220303T152000Z DTSTAMP:20260404T094120Z UID:MiniSymLattPoly22/4 DESCRIPTION:Title: Rational Ehrhart Theory\nby Sophie Rehberg (FU Berlin ) as part of Mini-Symposium on Lattice Polytopes\n\n\nAbstract\nThe Ehrhar t quasipolynomial of a rational polytope $P$ encodes fundamental arithmeti c data of $P$\, namely\, the number of integer lattice points in positive integral dilates of $P$. Ehrhart quasipolynomials were introduced in the 1 960s. They satisfy several fundamental structural results and have applica tions in many areas of mathematics and beyond. The enumerative theory of l attice points in rational (equivalently\, real) dilates of rational polyto pes is much younger\, starting with work by Linke (2011)\, Baldoni-Berline -Koeppe-Vergne (2013)\, and Stapledon (2017). We introduce a generating-fu nction ansatz for rational Ehrhart quasipolynomials\, which unifies severa l known results in classical and rational Ehrhart theory. In particular\, we define y-rational Gorenstein polytopes\, which extend the classical not ion to the rational setting. This is joint work with Matthias Beck and Sop hia Elia.\n LOCATION:https://stable.researchseminars.org/talk/MiniSymLattPoly22/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Akihiro Higashitani (Osaka) DTSTART:20220304T080000Z DTEND:20220304T085000Z DTSTAMP:20260404T094120Z UID:MiniSymLattPoly22/5 DESCRIPTION:Title: Lattice polytopes with small numbers of facets arising fr om combinatorial objects\nby Akihiro Higashitani (Osaka) as part of Mi ni-Symposium on Lattice Polytopes\n\n\nAbstract\nThere are several kinds o f lattice polytopes arising from combinatorial objects\, e.g.\, order poly topes\, chain polytopes\, edge polytopes\, matroid polytopes\, and so on. In this talk\, we introduce some of them and discuss when those families c oincide 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 o n the joint work with Koji Matsushita.\n LOCATION:https://stable.researchseminars.org/talk/MiniSymLattPoly22/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Rainer Sinn (Leipzig) DTSTART:20220304T104000Z DTEND:20220304T113000Z DTSTAMP:20260404T094120Z 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\nAbstra ct\nWe discuss unimodality of the $h^*$-vector for alcoved lattice polytop es (of Lie type $\\mathsf{A}$). The main ingredient is a fairly explicit t riangulation for which we need the assumption that the facets have lattice distance one to the set of interior lattice points. This is joint work wi th Hannah Sjöberg.\n LOCATION:https://stable.researchseminars.org/talk/MiniSymLattPoly22/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Marie-Charlotte Brandenburg (MPI MiS) DTSTART:20220304T130000Z DTEND:20220304T135000Z DTSTAMP:20260404T094120Z UID:MiniSymLattPoly22/7 DESCRIPTION:Title: Competitive Equilibrium and Lattice Polytopes\nby Mar ie-Charlotte Brandenburg (MPI MiS) as part of Mini-Symposium on Lattice Po lytopes\n\n\nAbstract\nThe question of existence of a competitive equilibr ium 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 b idder has 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 trans lated to a condition on certain lattice points in a lattice polytope.\n\nI n this talk\, we discuss this translation to the polyhedral language. Usin g polyhedral methods\, we show that in the case of the complete graph a co mpetitive equilibrium is indeed guaranteed to exist.\nThis is joint work w ith Christian Haase and Ngoc Mai Tran.\n LOCATION:https://stable.researchseminars.org/talk/MiniSymLattPoly22/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Francisco Santos (Cantabria) DTSTART:20220304T143000Z DTEND:20220304T152000Z DTSTAMP:20260404T094120Z UID:MiniSymLattPoly22/8 DESCRIPTION:Title: Empty simplices of large width\nby Francisco Santos ( Cantabria) as part of Mini-Symposium on Lattice Polytopes\n\n\nAbstract\nT he "flatness theorem” states that the maximum lattice width among all ho llow convex bodies in $\\mathbb{R}^d$ is bounded by a constant $\\operator name{Flt}(d)$ depending solely on $d$. For general $K$ the best current b ound is $Flt(d) \\le O(d^{4/3})$ (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 conv ex bodies of width more than linear is known.\n\nWe show two constructions leading to the first known $\\text{\\it empty simplices}$ (lattice simple x in which vertices are the only lattice points) of width larger than thei r dimension:\n\n• We introduce $\\text{\\it cyclotomic simplices}$ and e xhaustively compute all the cyclotomic $10$-simplices of volume up to $2^{ 31}$.\nAmong them we find five empty ones of width $11$\, and none of larg er width.\n\n• Using $\\text{\\it circulant}$ matrices of a specific for m\, we construct empty $d$-simplices of width growing asymptotically as $ d/\\operatorname{arcsinh}(1) \\sim 1.1346\\\,d$.\n\nThis is joint work wit h Joseph Doolittle\, Lukas Katthän and Benjamin Nill. See arXiv:2103.1492 5 for details.\n LOCATION:https://stable.researchseminars.org/talk/MiniSymLattPoly22/8/ END:VEVENT END:VCALENDAR