# Math

Below is a partial list of the math research I’ve done.

### Length-\(k\) Steenrod Operations

In Summer 2013 I worked on determining the structure of the subalgebras \(L(k)\) of the Steenrod Algebra as modules over the subalgebras \(\mathcal{A}(r)\), mentored by Mark Behrens. The hope is to show that in a certain sense \(L(k)\), when considered as a module over \(\mathcal{A}(r)\), is built up out of copies of the “quotient” algebras \(\mathcal{A}(r)//\mathcal{A}(r-k)\). I’m still working on writing up the results, but until then, you can see the poster I presented at the 2014 Joint Mathematics Meetings.

### Transposition Trees

In Summer 2012 at the University of Minnesota Duluth REU I worked on algorithms to find the diameters of groups generated by transposition trees. The idea is to consider the Cayley graph \(\Gamma\) of some permutation group as generated by some minimal set of transpositions, and find the diameter. In general finding the diameter of a Cayley graph is a very hard problem, but in this case we can get some good bounds; my research was on improving those bounds and the speed of the algorithms to compute them. The paper was published in Discrete Applied Mathematics.

### Elements of Random Matrices

At RSI I worked on a project in random matrices with Gregory Minton. The idea was to compute some properties of the distributions of the entries of random matrices in \(U(n)\). The paper is here and you can see my talk here.

### The Collatz Conjecture

In Summer 2009 Keenan Monks and I worked on understanding some properties of the Collatz map. The idea is that if you take \(\mathbb{Q}\) or the 2-adics and iterate the Collatz map, you get some directed graph, which has interesting symmetries; we were trying to understand those symmetries. The paper was published in Discrete Mathematics.