- (with Jenna Rajchgot) "Type A quiver loci and Schubert varieties", submitted, preprint available at arXiv:1307.6261.
We identify the representation space of an arbitrarily oriented type A quiver with an open subscheme of a Schubert variety, up to a smooth factor, with all orbit closures being given by Schubert conditions. Consequently, the space has a Frobenius splitting which compatibly splits the orbit closures, and we can realize the orbit closure poset as a subposet of the symmetric group under Bruhat order. This also gives an alternative proof of Bobiński and Zwara's result that these orbit closures are normal, Cohen-Macaulay, and have rational singularities.

- (with Calin Chindris and Jerzy Weyman) "Module varieties and representation type of finite-dimensional algebras", to appear in
*Int. Math. Res. Not.*, preprint available at arXiv:1201.6422.In this paper we seek invariant-theoretic characterizations of (Schur-)representation finite algebras. To this end, we introduce two classes of finite-dimensional algebras: those with the dense-orbit property and those with the multiplicity-free property. We show first that when a connected algebra A admits a pre-projective component, each of these properties is equivalent to A being representation-finite. Next, we give an example of a representation-infinite algebra with the dense-orbit property. We also show that the string algebras with the dense orbit-property are precisely the representation-finite ones. Finally, we show that a tame algebra has the multiplicity-free property if and only if it is Schur-representation-finite.

- "Tree modules and counting polynomials",
*Algebras and Representation Theory*16(5):1333-1347, 2013, preprint available at arXiv:1112.4782.We give a formula for counting tree modules for the quiver S_g with g loops and one vertex in terms of tree modules on its universal cover. This formula, along with work of Helleloid and Rodriguez-Villegas, is used to show that the number of d-dimensional tree modules for S_g is polynomial in g with the same degree and leading coefficient as the counting polynomial A_{S_g}(d, q) for absolutely indecomposables over F_q, evaluated at q=1.

- (with Ralf Schiffler) "Idempotents in representation rings of quivers",
*Algebra & Number Theory*6(5):967-994, 2012, preprint available at arXiv:1009.0029.We give a general procedure for taking a collection of directed graphs and maps between them to construct idempotents in representation rings, then show how to combinatorially orthogonalize them using a categorical version of Möbius inversion. This is applied to a category of directed graphs which gives orthogonal idempotents associated to all representations of a fixed quiver which are projective or injective on some subquiver. For more detail, see notes from 2010 Auslander conference below.

- "Rank Loci in Representation Spaces of Quivers", preprint available at arXiv:1004.1981.
Rank functions for quivers studied below give constructible functions on representation spaces of quivers, quiver Grassmannians, and certain moduli spaces. This is a geometric complement to my previous study of algebraic and combinatorial properties of rank functions.

- "New Inequalities for Subspace Arrangements",
*J. Combin. Theory Ser. A*, 118(1):152-161, 2011. Preprint available at arXiv:0905.1519.

An infinite hierarchy of inequalities that hold for the rank function of a subspace arrangement is given. Previously, only one non-elementary inequality for subspace arrangements was known. It gives new criteria for realizability (linear representability) of matroids. (

*See also the work of Dougherty, Freiling, and Zeger: arxiv:0910.0284.*) - Rank Functors and Representation Rings of Quivers (Ph.D. thesis), University of Michigan, 2009.
This essentially a concatenation of the two papers below, improved with the benefit of hindsight, more readers, and lack of space limitations (more examples, more background, and remarks on generalizations).

- "Rank Functions on Rooted Tree Quivers",
*Duke Math. J.*, 152(1):27-92, 2010. Preprint available at arXiv:0807.4496.The tools constructed in the paper below are combined with combinatorial methods, involving the category of quivers over a given quiver, to find structure in representation rings.

See also - "The Rank of a Quiver Representation",
*J. Algebra*, 320(6):2363-2387, 2008. Preprint available at arXiv:0711.1135.A functor is constructed that generalizes the rank of a linear map to the setting of an arbitrary diagram of vector spaces and linear maps (a quiver representation). Using maps of directed graphs, we get more, similar functors. These can be used to construct numerical invariants of a quiver representation which include, as the simplest cases, its dimension vector and the ranks of all maps appearing in the representation.

- Slides from ICRA 2012 in Bielefeld on modules varieties with dense orbits in every component and generic representation theory of algebras.
- Notes from a short talk on the tensor products of quiver representations and forbidden minors for MFT (AMS Syracuse, Fall 2010).
- Notes from SIGMA seminar expository talk on polymatroids and subspace arrangements.
- Notes from my talk at the 2010 Auslander Conference on work with Ralf Schiffler on tensor products of quiver representations.
- Slides from my talk at the Spring 2009 AMS Central Sectional Meeting. Basically a summary of my thesis.
- Here is a "wordle" made from the LaTeX code for my first paper! (It's not intended as a substitute for reading the original paper.)
- Notes for a talk on Gabriel's theorem (with a more combinatorial perspective) in the UM Student Representation Theory seminar, 2008-02-20.

**Notes from Math 5210 Introduction to Representation Theory and Lie Algebras** (University of Connecticut, Spring 2010)

Thanks to Ben Salisbury for typing these up during class (he has many more notes on his homepage). They also include some student presentations. To my knowledge, they haven't been carefully proofread.

**Some fractals** that I made as part of undergrad summer research with Estela Gavosto at the University of Kansas. These are (complex) one-dimensional slices of a (complex) two-dimensional parameter space arising from the Hénon map. The classical Mandelbrot set is, for example, one of the one-dimensional slices of this set, thus the similarity to some of these slices.