# 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.