Research Interests:

My research interests are in algebraic, geometric, and topological combinatorics. Some of my interests include:

• Lattice polytopes
• Graded semi-group algebras
• Polynomials arising from enumerative problems
• Topological properties of partially ordered sets
• Topological obstructions to graph colorings
• Simplicial complexes arising via combinatorial constructions

Funding: I am partially supported by the National Science Foundation via award DMS-0758321.

7. Independence Complexes of Stable Kneser Graphs, submitted.
   • For integers n >= 1, k >= 0, the stable Kneser graph SG(n,k) (also called the Schrijver graph) has as vertex set the stable n-subsets of [2n + k] and as edges disjoint pairs of n-subsets, where a stable n-subset is one that does not contain any 2-subset of the form {i, i + 1} or {1, 2n + k}. The stable Kneser graphs have been an interesting object of study since the late 1970's when A. Schrijver determined that they are a vertex critical class of graphs with chromatic number k+2. This article contains a study of the independence complexes of SG(n,k) for small values of n and k. Our contributions are two-fold: first, we find that the homotopy type of the independence complex of SG(2,k) is a wedge of spheres of dimension two. Second, we determine the homotopy types of the independence complexes of certain graphs related to SG(n,2).
   
   
6. Nowhere-Harmonic Colorings of Graphs, (joint with Matthias Beck), submitted.
   • Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley's theorem relating negative evaluations of the chromatic polynomial to acyclic orientations. Further, we discuss some examples demonstrating that nowhere-harmonic colorings are more complicated from an enumerative perspective than proper colorings.
   
   
5. Symmetries of the Stable Kneser Graphs, accepted to Advances in Applied Mathematics.
   • It is well known that the automorphism group of the Kneser graph KG(n,k) is the symmetric group on n letters, where KG(n,k) is the graph whose vertices are the k-subsets of an n-set and edges are given by disjoint k-sets. For n >= 2k + 1, k >= 2, we prove that the automorphism group of the stable Kneser graph SG(n,k) is the dihedral group of order 2n.
   
   
4. The Complex of Non-Crossing Diagonals of a Polygon, (joint with Richard Ehrenborg), accepted to Journal of Combinatorial Theory, Series A.
   • Given a convex n-gon P in the Euclidean plane, it is well known that the simplicial complex T(P) with vertex set given by diagonals in P and facets given by triangulations of P is the boundary complex of a polytope of dimension n-3. We generalize this result for any non-convex polygonal region P with n vertices and h+1 boundary components. We also provide a new proof that T(P) is a sphere when P is convex.
   
   
3. Ehrhart Polynomial Roots and Stanley's Non-negativity Theorem, (joint with Mike Develin)
   Integer Points in Polyhedra--Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, Contemporary Mathematics 2008, Volume: 452, pp 67-78.
   • Stanley's non-negativity theorem is at the heart of many of the results in Ehrhart theory. In this paper, we analyze the root behavior of general polynomials satisfying the conditions of Stanley's theorem and compare this to the known root behavior of Ehrhart polynomials. We provide a possible counterexample to a conjecture of the second author, M. Beck, J. De Loera, J. Pfeifle, and R. Stanley, and contribute some experimental data as well.
   
   
2. Norm Bounds For Ehrhart Polynomial Roots,
   Discrete and Computational Geometry, 39 (2008), no. 1-3, 191-193.
   • M. Beck, J. De Loera, M. Develin, J. Pfeifle and R. Stanley found that the roots of the Ehrhart polynomial of a d-dimensional lattice polytope are bounded above in norm by 1+(d+1)!. We provide an improved bound which is quadratic in d and applies to a larger family of polynomials.
   
   
1. An Ehrhart Series Formula For Reflexive Polytopes,
   Electronic Journal of Combinatorics, 13, no. 1 (2006), N 15.
   • This paper contains a proof that the numerator of the Ehrhart series for the free sum of a reflexive polytope and another lattice polytope containing the origin is the product of the numerators for the Ehrhart series of the summands.

• Ehrhart Theory for Lattice Polytopes.
  PDF.
  (My thesis is a combination of the papers "Norm Bounds...," "An Ehrhart Series Formula...," and "Ehrhart Polynomial Roots..." shown above.)

My Expository Papers:

• Making The Discrete Continuous: How Combinatorics Becomes Topology.
  PDF, PS.
  (Note that the contact information on this paper is out of date.)