Title: Ehrhart polynomials with negative coefficients



Abstract: The Ehrhart polynomials of integral convex polytopes count integer points under dilations of the polytopes. In this talk, I will discuss the possible sign patterns of the coefficients of Ehrhart polynomials of integral convex polytopes. While the leading terms, the second leading terms and the constant of Ehrhart polynomials are always positive, the other terms aren't necessarily positive.  In fact, some examples of Ehrhart polynomials with negative coefficients were known before. For arbitrary dimension, I will describe a construction of Ehrhart polynomials with negative coefficients. 

Title: Polyhedral Problems in Combinatorial Convex Geometry

Abstract: Polyhedra play a special role in combinatorial convex geometry in the sense that they are both convex sets and combinatorial objects.  As such, a polyhedron can act as either the convex set of interest or the combinatorial object describing properties of another convex set.  We will examine two instances of polyhedra in combinatorial convex geometry, one exhibiting each of these two roles.  The first instance arises in the context of Ehrhart theory, and the polyhedra are the central objects of study.  We will examine the Ehrhart h*-polynomials of a family of lattice polytopes called the r-stable (n,k)-hypersimplices, providing some combinatorial interpretations of their coefficients as well as some results on unimodality of these polynomials.  The second instance arises in algebraic statistics, and the polyhedra act as a conduit through which we study a nonpolyhedral problem.  For a graph G, we study the extremal ranks of the closure of the cone of concentration matrices of G via the facet-normals of the cut polytope of G.  Along the way, we will discover that real-rooted polynomials are lurking in the background of all of these problems.  

Kristen will discuss some old and new combinatorial games in this presentation for her qualifying exam.

Title:  Deletion-Induced Triangulations

Abstract:   Let $d > 0$ be  a fixed integer and let $\A \subseteq \mathbb{R}^d$ be a collection of $n \geq d+2$ points which we lift into $\mathbb{R}^{d+1}$. Further let $k$ be an integer satisfying $0 \leq k \leq n-(d+2)$ and assign to each $k$-subset of the points of $\A$ a (regular) triangulation obtained by deleting the specified $k$-subset and projecting down the lower hull of the convex hull of the resulting lifting. Next, for each triangulation we form the characteristic vector outlined by Gelfand, Kapranov, and Zelevinsky by assigning to each vertex the sum of the volumes of all adjacent simplices. We then form a vector for the lifting, which we call the compound GKZ-vector, by summing all the characteristic vectors. Lastly, we construct a polytope $\Sigma_k(\A) \subseteq \mathbb{R}^{| \A |}$ by taking the convex hull of all obtainable compound GKZ-vectors by various liftings of $\A$, and note that $\Sigma_0(\A)$ is the well-studied secondary polytope corresponding to $\A$. We will see that by varying $k$, we obtain a family of polytopes with interesting properties relating to Minkowski sums, Gale transforms, and Lawrence constructions, with the member of the family with maximal $k$ corresponding to a zonotope studied by Billera, Fillamen, and Sturmfels. We will also discuss the case $k=d=1$, in which we can outline a combinatorial description of the vertices allowing us to better understand the graph of the polytope and to obtain formulas for the numbers of vertices and edges present.

Title: Representing discrete Morse functions with polyhedra



Abstract: Discrete Morse theory is a method of reducing a CW complex to a simpler complex with similar topological properties. Well-known approaches to this task are due to Banchoff, whose process involves embedding a polyhedron in Euclidean space and considering the projections of its vertices onto a straight line, and to Forman, whose process involves finding special functions from the face poset of a complex to the real numbers. In this talk, I will discuss a result by Bloch which gives a relationship between these two methods. In particular, given a discrete Morse function on a CW complex, there exists a corresponding polyhedral embedding of the barycentric subdivision of X such that the discrete Morse function and the projection of the vertices of the polyhedron onto a line give the same critical cells.



 

Title: An Introduction to Symmetric Functions, part II

Abstract: In this pair of talks, I will provide an overview of basic results regarding symmetric functions.  My goal will be to create a "road map" for anyone who is interested in reading more about these objects in Chapter 7 of Stanley's Enumerative Combinatorics, Volume 2 (if you have a copy and are interested, it might be helpful to bring it with you).  We will motivate the study of symmetric functions by interpreting them as generalizations of subsets and multisubsets of [n], so these talks should be accessible to anyone who is familiar with the material from the first part of MA 614.

Title: An Introduction to Symmetric Functions, part I

Abstract: In this pair of talks, I will provide an overview of basic results regarding symmetric functions.  My goal will be to create a "road map" for anyone who is interested in reading more about these objects in Chapter 7 of Stanley's Enumerative Combinatorics, Volume 2 (if you have a copy and are interested, it might be helpful to bring it with you).  We will motivate the study of symmetric functions by interpreting them as generalizations of subsets and multisubsets of [n], so these talks should be accessible to anyone who is familiar with the material from the first part of MA 614.

Title: Single Splitter Details

Abstract: Lee defined the winding number w_k in a Gale diagram corresponding to a given simplicial polytope. He showed that w_k equals g_k of the corresponding polytope. We are working on extending Lee's definition of w_k to nonsimplicial polytopes. In this talk, we will discuss our results when the origin in the Gale diagram falls on a single k-splitter, a hyperplane that separates k points from the rest.

Title:  The polytope of Tesler matrices

Abstract:  Tesler matrices are upper triangular matrices with nonnegative integer entries with certain restrictions on the sums of their rows and columns. Glenn Tesler studied these matrices in the 1990s and in 2011 Jim Haglund rediscovered them in his study of diagonal harmonics. We investigate a polytope whose integer points are the Tesler matrices. It turns out that this polytope is a flow polytope of the complete graph thus relating its lattice points to vector partition functions. We study the face structure of this polytope and show that it is a simple polytope. We show its h-vector is given by Mahonian numbers and its volume is a product of consecutive Catalan numbers and the number of Young tableaux of staircase shape.  This is joint work with Brendon Rhoades and Karola Mészàros.