Masters Exam

Speaker:  Chloe Napier

Title:  TBA

Abstract:

Masters Exam

Speaker:  Chloé Napier, University of Kentucky

Title:     New Interpretations of the Two Higher Stasheff-Tamari Orders

Abstract:

In 1996, Edelman and Reiner defined the two higher Stasheff-Tamari orders on triangulations of cyclic polytopes and conjectured that they are equal. In 2021, Nicholas Williams defined new combinatorial interpretations of these two orders to make the definitions more similar. He builds upon the work by Oppermann and Thomas in the even dimensional case of giving an algebraic analog to these orders using higher Auslander-Reiten Theory. He then gives a completely new result for the odd dimensional case. In this talk, we will discuss the combinatorial interpretations of the even dimensional case and motivate the odd dimensional case and algebraic analog by example. 

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.

Title:  Not all Simplicial Polytopes are Weakly Vertex-Decomposable

Abstract:  The Simplex Method solves linear programs by testing adjacent vertices in the feasible set (a polytope) in sequence such that each new vertex in the sequence improves or stays the same with respect to the objective function. It is natural to ask how long the Simplex Method could take to solve a given linear program. Stating this in the language of polytopes, we would like to find a bound on the diameter of d-dimensional polytopes with a fixed number of vertices, say n. In 1980, Billera and Provan defined the notions of k-decomposability and weak k-decomposability for simplicial complexes and computed bounds on the diameter of complexes admitting such decompositions. In particular, these bounds become linear in n and d when k=0. Hence, it is reasonable to ask if all simplicial complexes admit such a decomposition. In 2010, De Loera and Klee identified a simple transportation polytope in each dimension greater than three whose dual polytope is not weakly vertex-decomposable. We will introduce the notion of weak k-decomposability and transportation polytopes, and then we will see why the family of polytopes constructed by De Loera and Klee fail to be weakly vertex-decomposable.

In this talk, we will discuss part of Richard Stanley's paper "Generalized Riffle Shuffles and Quasisymmetric Functions". After defining the QS-distribution by standardizing elements from the probability distribution on a totally ordered set, we will examine another description in terms of riffle shuffles. Then we will consider the relationship between the QS-distribution and quasisymmetric functions. Using this relationship, we can generalize results concerning quasisymmetric functions and symmetric functions.