# 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.

The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota.