Sept 10 
Richard Ehrenborg
University of Kentucky 
Uniform flag triangulations of the Legendre polytope
(or how I spent my summer holiday)
The Legendre polytope, also known as the full root polytope of type A, is the convex hull of all pairwise differences of the basis vectors. We describe all flag triangulations of this polytope that are uniform, that is, the edges are described in terms of the relative order of the indices of the four basis vectors involved. We obtain three classes of triangulations: the lex class, the revlex class and the Simion class. We also do a refined enumeration of faces of these triangulations that keeps track of the number of forward and backward arrows, and surprisingly the enumeration result only depends on which class the triangulation belongs to. Joint work with Gabor Hetyei and Margaret Readdy. 
Sept 17 
Margaret Readdy
University of Kentucky 
Combinatorial identities related to the calculation of
the
cohomology of Siegel modular varieties
In the computation of the intersection cohomology of Shimura varieties, or of the L^{2} cohomology of equal rank locally symmetric spaces, combinatorial identities involving averaged discrete series characters of real reductive groups play a large technical role. These identities can become very complicated and are not always wellunderstood. We propose a geometric approach to these identities in the case of Siegel modular varieties using the combinatorial properties of the Coxeter complex of the symmetric group Joint work with Richard Ehrenborg and Sophie Morel. 
Oct 1 
Ben Braun
University of Kentucky 
Graph constructions and chromatic numbers
We will survey various graph constructions related to proper kcolorability. Possible topics for discussion include constructions of kchromatic graphs due to Hajos, Ore, and Urquhart, constructions of graphs with high girth and high chromatic number due to Alon, Kostochka, Reiniger, West, and Zhu, and constructions of kcritical graphs with minimal possible number of edges due to Kostochka and Yancey.

Oct 8 
Matthias Köppe
UC Davis 
Normalized antics: Polyhedral computation in an irrational age
Classic polyhedra such as the dodecahedron have irrational coordinates. How do we compute with them if we need exact answers? In this handson lecture using the SageMath system, intended to be accessible to advanced undergraduate students, I explain how to compute with polyhedral representations and with real embedded algebraic number fields. I then report on an ongoing project with W. Bruns, V. Delecroix, and S. Gutsche to obtain an efficient implementation in Normaliz, integrated with SageMath, involving the existing software libraries ANTIC, arb, and FLINT.

Oct 15

Alex Chandler
NC State 
Thin posets and homology theories
Inspired by BarNatan's description of Khovanov homology, we discuss thin posets and their capacity to support homology and cohomology theories which categorify rankstatistic generating functions. Additionally, we present two main applications. The first, a categorification of certain generalized Vandermonde determinants gotten from the Bruhat order on the symmetric group by applying a special TQFT to smoothings of torus link diagrams. The second is a broken circuit model for chromatic homology, categorifying Whitney's broken circuit theorem for the chromatic polynomial of graphs.

Oct 22  No meeting 
Robert Lang Origami workshop 25 pm

Oct 29 
Ricky Liu
NC State 
Ppartition generating functions of naturally labeled posets
The Ppartition generating function of a (naturally labeled) poset P is a quasisymmetric function enumerating orderpreserving maps from P to the positive integers. We give several necessary and sufficient conditions for when two posets can have the same Ppartition generating function. We also show that the Ppartition generating function of a connected poset is an irreducible element of the ring of quasisymmetric functions. The proofs utilize the Hopf algebra structure of posets and quasisymmetric functions. This is joint work with Michael Weselcouch.

Nov 5  No meeting 
HaydenHoward Lecture 4 pm

Nov 12 
Yuan Zhou
U Kentucky 
Integer optimization, cutting planes, and approximation theory
Cutting planes are the workhorses of numerical integer optimization. In my talk, I review the principles of the leading approach to solving integer linear optimization problems. I then introduce my research on the theory of generalpurpose cutting planes. I end the talk with a recent result regarding the approximation theory of socalled cutgenerating functions in a particular model, Gomory and Johnson's infinite group problem. Our approximation theorem has nice "injective" properties, which have implication on the relation between socalled finite group relaxations and the infinite group problem.

Nov 19 
Fernando Shao
U Kentucky 
Maximizing the number of solutions to a linear equation
When two numbers are added in base 10, what's the chance that a carry occurs? Questions like this motivates the study of maximizing/minimizing the number of solutions to a linear equation, such as x+y=z, with the variables lying in a set of given size N. Some of these questions are easy (i.e. challenge problems for high school students), but some others are hard (i.e. open). I will discuss problems from both categories. Based on joint work with P. Diaconis and K. Soundararajan.

Nov 26 
Tefjol Pllaha
U Kentucky 
Laplacian Simplices II: A Coding Theoretic Approach
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.

Dec 3 
Matias von Bell
U Kentucky 
The ASMCRY(n) Polytope: An Interesting Face of the Alternating Sign Matrix Polytope
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 ChanRobbinsYuen polytope (ASMCRY(n)). It is part of a larger family: The ASMCRY family of polytopes. After a discussion of flow and order polytopes, we'll prove that the polytopes in the ASMCRY 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.

Dec 6 
Susanna Lang
U Kentucky 
Rational Catalan Numbers and Associahedra
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.

Jan 24 
Discrete Math Job Candidate


Jan 29 
Discrete Math Job Candidate


Feb 1 
Discrete Math Job Candidate


Feb 25 
McCabe Olsen
Ohio State 
Signed Birkhoff polytopes and the orthantlattice preservation property
Given a ddimensional lattice polytope P, we say that P has the orthantlattice 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 typeB 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). 
Mar 7 
Gabor Hetyei
UNC Charlotte 
Alternation acyclic tournaments and the homogeneous Linial arrangement
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. 
Mar 8 
Galen DorpalenBarry
U Minnesota 
Whitney Numbers for Cones
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. 
Last updated March 4, 2019.