I am a graduate student at the University of Minnesota. My research interests are in algebraic combinatorics. My advisor is Victor Reiner. Currently, I am studying matroids and vector configurations by associating group representations to them. Here is my CV.
[ps, pdf] Products of Linear Forms and Tutte Polynomials. This paper studies the vector space spanned by products of linear forms from a fixed set Δ. Using a result of Orlik and Terao we obtain a doubly indexed direct sum of this space. The main theorem is that the resulting Hilber series is the Tutte polynomial evaluation T(Δ;1+x,y). By specializing x and y we obtain various results from the literature.
[ps, pdf] A short proof of Gamas's Theorem (Submitted). This is a short and self-contained proof of Gamas's Theorem on the vanishing of symmetrized tensors.
Here is a list of talks I have given.
Representations generated by tensors (4-15-2008). This talk was given to the Combinatorics Seminar at the University of Minnesota. I talked about the symmetric and general linear group group representations generated by a single tensor. I will give a version of this talk in October 2008 at an AMS special session on combinatorial representation theory.
A symmetric function for realized matroids (4-21-2007). This brief talk is about vanishing of symmetry classes of tensors. Their vanishing is known to depend only on a matroid. We introduce a symmetric function that appears useful in studying these symmetrizations and sketch a proof of a theorem due to Merris.
Representation Theory of sl(n) (4-2-2007). I spoke to some of my fellow students about the translation of the study of finite dimensional irreducible sl(n) modules to a problem in multilinear algebra.
&Delta operators and their basic polynomials (3-9-2007). Another talk to the Student Combinatorics group at the University of Minnesota. I talked about using linear algebra to prove things about analouges of derivatives on the vector space of polynomials over C. The culmination of this is a simple proof of Cayley's theorem on the number of trees on n vertices.
Cyclic Group Actions, Jeu de Taquin and Cyclic Sieving (2-1-2007). I spoke to the undergraduate math club about the cyclic group action that promotion (a variant of Jeu de Taquin) gives rise to when played on standard Young tableuax of a fixed shape. An instance of the cyclic sieving phenomenon is mentioned.
Matroid Colorings (11-19-2006). This talk was given to the Student Combinatorics group here at Minnesota. I talked about matroids, matroid colorings and vanishing of symmetrized tensors.
Generalized Matrix Functions (9-12-2006). These are slides from the Junior Colloquium I gave on a satisfying way to generalize the determinant.
Eigenvalues of Graphs (10-30-2005). These are slides for the talk I gave at the Student Combinatorics Seminar on the spectra of graphs.
Here we have my undergraduate work on critical groups of graphs. The critical group of a graph is the torsion subgroup of the cokernel of the Laplacian matrix. The order of the critical group of a graph is the number of spanning forests in the graph.
[ps, pdf] Critical groups of some regular line graphs. This paper arose out of the summer 2003 REU program at the University of Minnesota. The critical group of the line graph of the complete bipartite graph is given. Conjectures are also presented that form the basis for future work were the critical group of a graph is related to that of its line graph.
[ps, pdf] Critical groups and line graphs. This was my undergraduate senior thesis. In it, a homomorphism is given relating the lattice of integer cycles of a line graph to that of the original graph. This homomorphism extends natuarly to the critical group when the original graph is regular. An exact sequence relating the critical group of the line graph to the critical group of the first edge subdivision is conjectured. (As of October 2005 all of the conjectures in this paper have been resolved positively by Berget, Maxwell and Reiner. Further work was done by Manion and Potechin. Finally, a preprint is available.)