DISCRETE CATS SEMINAR
Discrete CATS Seminar
Discrete CATS Seminar
Discrete CATS Seminar
Title: Algebraic Aspects of Lattice Simplices
Abstract: Given a lattice polytope P, there are open problems of interest related to the integer decomposition property, Ehrhart h*-unimodality, and Ehrhart positivity. In this talk, we will survey some recent results in this area, based on various joint works with Rob Davis, Morgan Lane, Fu Liu, and Liam Solus.
Discrete CATS Seminar
Title: Algebraic Aspects of Lattice Simplices
Abstract: Given a lattice polytope P, there are open problems of interest related to the integer decomposition property, Ehrhart h*-unimodality, and Ehrhart positivity. In this talk, we will survey some recent results in this area, based on various joint works with Rob Davis, Morgan Lane, Fu Liu, and Liam Solus.
Discrete CATS Seminar
Title: Mutation of friezes
Abstract: Frieze is an array of positive integers satisfying certain rules. Friezes of type A were first studied by Conway and Coxeter in 1970's, but they gained fresh interest in the last decade in relation to cluster algebras. Moreover, the categorification of cluster algebras developed in 2006 yields a new realization of friezes in terms of representation theory of Jacobian algebras. In this talk, we will discuss friezes of types A and D and their mutations.
Discrete CATS Seminar
Title: Mutation of friezes
Abstract: Frieze is an array of positive integers satisfying certain rules. Friezes of type A were first studied by Conway and Coxeter in 1970's, but they gained fresh interest in the last decade in relation to cluster algebras. Moreover, the categorification of cluster algebras developed in 2006 yields a new realization of friezes in terms of representation theory of Jacobian algebras. In this talk, we will discuss friezes of types A and D and their mutations.
Discrete CATS Seminar
Speaker: Gabor Hetyei, UNC Charlotte
Title: Alternation acyclic tournaments and the homogeneous Linial arrangement
Abstract:
We define a tournament to be alternation acyclic if it does not
contain a cycle in which descents and ascents alternate. We show that
these label the regions in a homogenized generalization of the Linial
arrangement. Using a result by Athanasiadis, we show that these are
counted by the median Genocchi numbers. By establishing a bijection
with objects defined by Dumont, we show that alternation acyclic
tournaments in which at least one ascent begins at each vertex, except
for the largest one, are counted by the Genocchi numbers of the first
kind. As an unexpected consequence, we obtain a simple model for the
normalized median Genocchi numbers.
Discrete CATS Seminar
Speaker: Galen Dorpalen-Barry, University of Minnesota
Title: Whitney Numbers for Cones
Abstract:
An arrangement of hyperplanes dissects space into connected components
called chambers. A nonempty intersection of halfspaces from the
arrangement will be called a cone. The number of chambers of the
arrangement lying within the cone is counted by a theorem of
Zaslavsky, as a sum of certain nonnegative integers that we will call
the cone's "Whitney numbers of the 1st kind". For cones inside the
reflection arrangement of type A (the braid arrangement), cones
correspond to posets, chambers in the cone correspond to linear
extensions of the poset, and these Whitney numbers refine the number
of linear extensions. We present some basic facts about these Whitney numbers,
and interpret them for two families of posets.
Discrete CATS Seminar
Speaker: McCabe Olsen, Ohio State University
Title: Signed Birkhoff polytopes and the orthant-lattice preservation property
Abstract:
Given a d-dimensional lattice polytope P, we say that P has the orthant-lattice preservation property (OLP) if the subpolytope obtained by restriction to any orthant is a lattice polytope. While this property feels somewhat contrived, it can actually be quite useful in verification of discrete geometric properties of P. In this talk, we will discuss a number of results for the existence of triangulation and the integer decomposition property for reflexive OLP polytopes. One such polytope which fits into the program is a type-B analogue of the Birkhoff polytope and its dual polytope, the investigation of which led to interest in this property. This is based on joint work with Florian Kohl (Aalto University).