Lattice polytopes, with a view towards Geometry and Applications

Titles and abstracts

• • Alex Fink (Queen Mary)
  • Newton polytopes of Schubert polynomials
  • We show that the support of a Schubert polynomial is the set of lattice points of a generalised permutohedron. The techniques extend to some related combinatorial polynomials from flag varieties, such as Demazure's key polynomials. We also characterise when a Schubert polynomial has all coefficients 0 or 1. This is joint work with Karola Mészáros and Avery St. Dizier.
• • Christian Haase (FU Berlin)
  • Newton-Okounkov functions - a toric case study
  • I will talk about joint work in progress with Alex Küronya and Lena Walter on Newton-Okounkov bodies. For a smooth projective toric variety X with an ample toric line bundle L on it and a toric flag Y_. in it, the Newton-Okounkov body P equals the moment polytope of L - a lattice polytope. The order of vanishing of sections of L in a point p \in X (or in any subvariety of X) induces a convex function on P. If p is a torus fixed point (or any torus invariant subvariety), this function is linear. If p is a point in the torus, the function is piece-wise linear and things get more interesting. As a Corollary we can show in many cases rationality of Seshadry constants.
• • Max Hlavacek (Berkeley)
  • Dehn-Sommerville equations for cubical polytopes
  • The classical Dehn-Sommerville equations, relating the face numbers of simplicial polytopes, have an analogue ​for cubical polytopes. We show how to obtain these relations by considering the zeta polynomial of the face lattices of cubical polytopes, as well as the Ehrhart polynomials of these polytopes. We also explore how these ideas connect to chain partitions of these face lattices.
• • Johannes Hofscheier (Nottingham)
  • The cohomology ring of toric bundles
  • Khovanskii and Pukhlikov observed that the cohomology ring of a smooth projective toric variety is completely described by the Bernstein-Kushnirenko theorem. In this talk, I will report on joint work with Khovanskii and Monin where we extend this approach to obtain a description of the cohomology ring of an equivariant compactification of a torus principal bundle by formulating respective Bernstein-Kushnirenko-type theorems.
• • Nathan Ilten (Simon Fraser)
  • Algebraic hyperbolicity for surfaces in toric threefolds
  • Given a Laurent polynomial f in 3 variables with Newton polytope P, what can be said about the geometric genera of curves lying on the surface V(f) in (\mathbb{C}^*)^3? In the case that P is the d-th dilate of a standard simplex and f is very general, Xu gives a lower bound on the geometric genus of any curve lying on V(f) in terms of d and its degree. In this talk, I will discuss joint work with Christian Haase in which we generalize this result to arbitrary smooth polytopes P. This leads to new insights into the algebraic hyperbolicity of surfaces in smooth toric threefolds.
• • Katharina Jochemko (KTH)
  • Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity
  • Generalized permutahedra form a combinatorially rich class of polytopes that naturally appear in many areas of mathematics such as combinatorics, geometry, optimization and statistics. They comprise many important classes of polytopes, for example, matroid polytopes. We study functions on generalized permutahedra that behave linearly with respect to dilation and taking Minkowski sums. We give a complete classification of all positive, translation-invariant Minkowski linear functionals on permutahedra that are invariant under permutations of the coordinates: they form a simplicial cone and we explicitly describe the generators. We apply our results to prove positivity of the linear coefficient of the Ehrhart polynomial of any generalized permutahedron thereby supporting a conjecture of Castillo and Liu (2018). This is joint work with Mohan Ravichandran.
• • Michael Joswig (TU Berlin)
  • Computing triangulations of lattice polytopes
  • Triangulations of finite point configurations are known to have numerous applications in many areas of mathematics, ranging from algebraic geometry and optimization to applications in physics, biology and beyond. The case where the given points form the vertices of a lattice polytope or the entire set of all lattice points in a polytope is particularly interesting. We will survey some recent results on computing specific triangulations as well as enumerating the entire set of all triangulations of various configurations of lattice points.
• • Elana Kalashnikov (Harvard)
  • Finding mirrors for quiver flag zero loci
  • One interesting feature of the classification of smooth Fano varieties up to dimension three is that they can all be described as certain subvarieties in GIT quotients; in particular, they are all either toric complete intersections (subvarieties of toric varieties) or quiver flag zero loci (subvarieties of quiver flag varieties). Fano varieties are expected to mirror certain Laurent polynomials; given such a Fano toric complete intersections, one can produce a Laurent polynomial via the Landau-Ginzburg model. In this talk, I'll discuss finding mirrors of four dimensional Fano quiver flag zero loci via finding degenerations of the ambient quiver flag varieties.
• • Oleg Karpenkov (Liverpool)
  • Global relations for toric singularities
  • In this talk we will discuss a link between geometry of continued fractions and global relations for singularities of complex projective toric surfaces. The results are based on recent development of lattice trigonometric functions that are invariant with respect to Aff(2,Z)-group action.
• • Dimitra Kosta (Glasgow)
  • Maximum likelihood estimation of toric Fano varieties motivated by phylogenetics
  • We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all 2-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two singular points of type A1 and provide explicit expressions that allow to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using A-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fibre products, and it also follows from a connection to a recent result about staged trees. This is joint work with Carlos Amendola and Kaie Kubjas.
• • Diane Maclagan (Warwick)
  • Higher connectivity of tropical varieties and a tropical Bertini theorem
  • Balinski's theorem states that the vertex-edge graph of a d-dimensional polytope is d-connected. More generally, the hypergraph with vertices the k-dimensional faces of P, and hyperedges corresponding to dimension k+1 faces, is (d-k)-connected. This is a corollary of a more general theorem, joint with Josephine Yu, on the higher connectivity of tropical varieties. A key part of the proof is a tropical Bertini Theorem, which in turn relies on a recent toric Bertini theorem of Fuchs, Mantova, and Zannier.
• • Fatemeh Mohammadi (Bristol)
  • Generalized permutohedra from probabilistic graphical models
  • Graphical models (Bayesian networks) based on directed acyclic graphs (DAGs) are used to model complex cause-and-effect systems. A graphical model is a family of joint probability distributions over the nodes of a graph which encodes conditional independence relations via the Markov properties. One of the fundamental problems in causality is to learn an unknown graph based on a set of observed conditional independence relations. In this talk, I will describe a greedy algorithm for DAG model selection that operates via edge walks on so-called DAG associahedra. For an undirected graph, the set of conditional independence relations are represented by a simple polytope known as the graph associahedron, which can be constructed as a Minkowski sum of standard simplices. For any regular Gaussian model, and its associated set of conditional independence relations we construct the analogous polytope DAG associahedron which can be defined using relative entropy. For DAGs we construct this polytope as a Minkowski sum of matroid polytopes corresponding to Bayes-ball paths in a graph. This is joint work with Caroline Uhler, Charles Wang, and Josephine Yu.
• • Benjamin Nill (Magdeburg)
  • From spanning lattice polytopes to the Gromov width
  • A lattice polytope is called spanning if its lattice points affinely span the lattice. This large class of lattice polytopes recently gained interest as they share some nice Ehrhart-theoretic properties with IDP polytopes. In this talk, I will give a motivating overview on these results (joint work with Johannes Hofscheier and Lukas Katthän). I will also report on some new observations regarding the lattice width of non-spanning lattice polytopes and its relation to the Gromov width for symplectic toric manifolds (joint work with Gennadiy Averkov and Johannes Hofscheier).
• • Hendrik Süß (Manchester)
  • Toric topology of G(2,5) and the del Pezzo surface of degree 5
  • This talk is concerned with the topology of the orbit space of the action of a maximal compact torus on the complex Grassmannian G(2,5). More precisely, I will discuss how to calculate the integral homology groups of this topological space. This will take us on a tour from the classical birational geometry of the del Pezzo surface of degree 5 to the more recent theme of Variation of Geometric Invariant Theory which in turn, by the Kempf-Ness Theorem, links to the topology of the orbits space of the compact torus action.
• • Stefano Urbinati (Udine)
  • Local chamber decomposition and compatible toric embeddings
  • In this seminar I will present the first part of a work in collaboration with Alex Küronya. In Hu-Keel's work on Mori Dream Spaces, the authors give a local version of their main theorem. However, this result, which was not proven, turns out to be false in full generality. In particular I will give a counterexample and introduce the concept of gen divisors, so as to be able to give a correct version. Thanks to this new version, we are able to give an immersion of the starting variety in a toric variety that manages to locally control all the operations of the Minimal Model Program.
• • Martin Vodička (Max Planck Institute)
  • Normality of Kimura 3-parameter model
  • Phylogenetic is the science that models evolution and one of the central objects is group-based tree model. To every tree and a group there is associated algebraic variety which is just (the closure of) the set of all possible probability distribution allowed by the model. It was proven that these varieties are toric and there is explicit description in terms of lattice polytopes. The most meaningful is 3-Kimura model which is group model with the underlying group Z_2\times Z_2. We show that the polytopes associated to this model are always normal which implies projective normality of associated toric varieties.