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

Title:   Laplacian Simplices II: A Coding Theoretic Approach

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

Abstract: 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.

Speaker:  Yuan Zhou, U Kentucky

Title:  Integer optimization, cutting planes, and approximation theory

Abstract: 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 general-purpose cutting planes. I end the talk with a recent result regarding the approximation theory of so-called cut-generating 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 so-called finite group relaxations and the infinite group problem.

Speaker:  Ricky Liu, NC State

Title:  P-partition generating functions of naturally labeled posets

The P-partition generating function of a (naturally labeled) poset P is a quasisymmetric function enumerating order-preserving maps from P to the positive integers. We give several necessary and sufficient conditions for when two posets can have the same P-partition generating function. We also show that the P-partition 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.

Speaker:  Alex Chandler, NC State

Title:  Thin posets and homology theories

Abstract:

Inspired by Bar-Natan's description of Khovanov homology, we discuss

thin posets and their capacity to support homology and cohomology

theories which categorify rank-statistic 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.

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