# Research

My broad areas of research are computational algebra and algebraic geometry. A summary of my research experience and goals can be read in my **Research Statement**.

# Reesearch Publications

**Sums-of-Squares Formulas over Algebraically Closed Fields**

Journal of Algebra, Volume 497, 1 March 2018, Pages 393-410.

I reformulate the existence of sums-of-squares formulas as a question in algebraic geometry, and I prove that, for large enough p, existence of sums-of-squares formulas over algebraically closed fields is independent of the characteristic, among other results.**A Constraint Propagation Algorithm for Sums-of-Squares Formulas over the Integers**

In preparation

I introduce an efficient algorithm for finding sums-of-squares formulas over the integers, discuss how combinatorial arguments can be used to further improve its run-time, and demonstrate its effectiveness for large formulas. Preprint is available**here**.**Sums-of-Squares Formulas over Z/mZ**

In preparation

I provide a new representation of sums-of-squares formulas over Z/mZ, analogous to the representation of sums-of-squares formulas over the integers as consistently signed intercalate matrices. Using this formulation, I introduce an algorithm for finding sums-of-squares formulas over Z/mZ, and demonstrate how this algorithm can be used to produce sums-of-squares formulas which do not lift to the integers.**Sums-of-Squares Formulas over Arbitrary Fields**

Ph.D. Thesis, 2016

Available**here**.

# Supervised Undergraduate Research

I have supervised the following undergraduate research projects:

- Signatures and Canonical Representations of Sums-of-Squares formulas over the Integers.
- Orbits of Sums-of-Squares formulas over the Integers.
- Algebraic Number Theory.
- Algebraic Topology.
- Knot Theory.

I have also supervised the following senior projects:

- Elliptic Curves.
- Option Pricing.
- Linear Cryptanalysis on a simple SPN.
- Cryptography and Secret Sharing in Video Game Design.
- Applications of Mathematics to Electrical Engineering.