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.

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).

Speaker:  Susanna Lang, U Kentucky

Title:  Rational Catalan Numbers and Associahedra

Abstract:

Classical Catalan numbers are known to count over 200 combinatorial objects, including Dyck paths, noncrossing partitions, and vertices of the classical associahedra. In this talk we discuss a generalization of the classical Catalan numbers and their connection with a class of simplicial complexes known as rational associahedra. We show rational associahedra have many nice properties, in particular they are shellable. This talk follows the paper "Rational Associahedra and Noncrossing Partitions" by Armstrong, Rhoades, and Williams.

This is Susanna Lang's Masters exam.

Speaker:  Matias von Bell

Title:  The ASMCRY(n) Polytope: An Interesting Face of the Alternating Sign Matrix Polytope.

Abstract:
This talk showcases some of the work done by Karola Meszaros, Alejandro Morales, and Jessica Strikerin their 2015 paper titled ``On Flow Polytopes, Order Polytopes, and Certain Faces of the Alternating Sign Matrix Polytope".
The alternating sign matrix polytope is the convex hull of alternating sign matrices. We will focus on one of its faces known as the Alternating Sign Matrix Chan-Robbins-Yuen polytope (ASMCRY(n)). It is part of a larger family: The ASM-CRY family of polytopes. After a discussion of flow- and order polytopes, we'll prove that the polytopes in the ASM-CRY family are both flow- and order polytopes. Then using Stanley's results for order polytopes, we show that ASMCRY(n) has Catalan many vertices, and its volume is the number of Standard Young Tableaux of staircase shape.

This is Matias von Bell's Masters Exam.

Speaker:  Tefjol Pllaha, U Kentucky

Abstract:

This talk will be about Laplacian simplices, that is, simplices whose vertices are rows of the Laplacian matrix of a simple connected graph. We will focus on graphs and graph operations that yield reflexive Laplacian simplices. We spot such graphs by showing that the $h^*$-vector of the simplex is symmetric. We use the same approach as Batyrev and Hofscheier by considering the fundamental parallelepiped lattice points as a finite abelian group. This is joint work with Marie Meyer.

www.math.uky.edu/~readdy/Seminar

Speaker:  Fernando Shao, U Kentucky

Title Maximizing the number of solutions to a linear equation

Speaker:  Yuan Zhou, U Kentucky

Title:  Integer optimization, cutting planes, and approximation theory

Speaker:  Ricky Liu, NC State

Title:  P-partition generating functions of naturally labeled posets