I am a fifth-year PhD student in the Department of Mathematics at the University of Michigan, Ann Arbor.
I graduated in 2017 from the University of Washington, Seattle, with a B.S. in mathematics and a minor in music.

## Contact info

Email: willdana at umich dot edu
Office: East Hall 3852

## Research

I am broadly interested in combinatorics and representation theory. Specifically, I'm currently studying relationships between Coxeter groups, quiver representations, and representations of preprojective algebras. My advisor is David Speyer.
I've also looked at Brascamp-Lieb inequalities, integral inequalities associated to certain quiver representations, with Harm Derksen.
More broadly, I enjoy any math that involves pictures with arrows in them. Combinatorics and representation theory both fit the bill.

## Writing

• "Stability of stretched root systems, root posets, and shards." The Electronic Journal of Combinatorics, 28(3), 2021.

One thing that almost all infinite families of finite and affine Coxeter groups have in common is that their Coxeter diagrams are obtained by inserting a long path into a fixed diagram. This paper looks at this "insert a long path" construction for any diagram, and shows how some related constructions (the root poset and the arrangement of shards) stabilize. The original inspiration for this came from wanting a nice uniform description of the bricks of type D preprojective algebras, though this paper is purely combinatorial.

• With David Jekel, Avi Levy, Austin Stromme, and Collin Litterell: "Algebraic Properties of Generalized Graph Laplacians: Resistor Networks, Critical Groups, and Homological Algebra." SIAM J. Discrete Math., 32(2), 1040--1110, 2018. Preprint: arXiv:1604.07075.

This came out of the 2015 REU at the University of Washington. The basic idea is to build up a general foundation for talking about the sandpile group or critical group of a graph, allowing for coefficient rings other than Z, and to connect this to a bunch of other things. I wasn't directly involved with much of this paper, but I computed a lot of examples and worked on the material that became section 7.

• "Seeing Galois Theory on Riemann Surfaces with Dessins d'Enfants." 2017.

My senior thesis, an exposition on dessins d'enfants. Written under the guidance of Professor Jim Morrow. The plan was to explain the arguments in Ernesto Girondo and Gabino GonzĂˇlez-Diez's Introduction to Compact Riemann Surfaces and Dessins d'Enfants in a way I could understand better.

• With David Jekel: A note on critical groups of graphs with dihedral symmetry.

The number of spanning trees of an undirected Cayley graph of the cyclic group of order n is always n times a perfect square. This note arose out of trying to find a nice algebraic explanation of this fact using sandpile groups.

## Teaching

### UM

Fall 2017: Math 105.
Winter 2018: Math 115.
Fall 2018: Math 116.
Winter 2019: Math 116.
Fall 2021: Math 105.
(the latter third of) Winter 2022: Math 115.

### UW

At the University of Washington, I was the TA for the second-year honors calculus sequence in 2015-2016 (Math 334, 335, 336) and 2016-2017 (Math 334, 335, 336).

### Mathcamp

In summer 2019, I was a mentor at Canada/USA Mathcamp. I taught classes on complex projective space, ring theory, quiver representations, and determinants, as well as two- and one-day classes on random spanning trees and matroids.
Here are notes for my class on quiver representations.
Here are notes for my class on determinants.
Here is a note on why the groundskeeper's algorithm for generating uniformly random spanning trees works.
Here are notes on my one-day class on matroids.
Please let me know if you find typos.