The roots of my mathematical research are in computational number theory. I've spent a lot of time thinking about various finiteness results related to the representation problem for quadratic lattices. These types of problems are the basis of some of the most classical questions in number theory, for example, what sort of positive integers can be represented by a sum of three squares? Many of these questions date back to antiquity, and their resolution has only become tractable in the recent decades. Accordingly, my approach to these problems involves a melding of the classical geometry of quadratic lattices and the algebraic structures associated to this geometry with modern computational tools like SageMath, Magma, and the LMFDB.
My interest in algorithmic and data-driven number theory naturally led me to explore more applicable uses for our current computational capabilities. I recently spent a year working in a data science group with a Boston area based start-up, Tagup, which sparked my interest in machine learning algorithms. Lately, I've been spending a great deal of time considering how methods of sampling and variational inference can be used to analyze complicated multi-layered timeseries data.
One of my favorite things about data science is the diversity of backgrounds that are attracted to the field and the unique approaches and capabilities to problem solving that this brings.
Here are some links to various talks I've given:
- Spinor Regular Ternary Quadratic Lattices, Institute for Computational and Experimental Research in Mathematics, April 18 (ICERM archive video)
- Almost Universal Ternary Sums of Polygonal Numbers, Southern Illinois University Carbondale, November 2017 (YouTube video).
- (with M. Dutour Sikiríc, J. Voight and W. van Woerden) A canonical form for positive definite matrices, Proceedings of the Fourteenth Algorithmic Number Theory Symposium, ANTS-XIV, Mathematical Sciences Publishers, 2020 (final version at arXiv:2004.14022).
- (with A. G. Earnest) Classification of one-class spinor genera for quaternary quadratic forms,Act Arith. 91 3 (2019) 259-287, (final version at arXiv:1803.03028).
- (with B. Kane) An algebraic and analytic approach to spinor exceptional behavior in translated lattices, Automorphic Forms and Related Topics, Contemp. Math., 732, 2019, Amer. Math.
Soc., Providence, RI, (final version pdf available here).
- (with A. G. Earnest) Completeness of the list of spinor regular ternary quadratic forms, Mathematika, 65 (2019), 213-235, (final version at arXiv:1711.05811).
- (with B. Kane) Almost universal ternary sums of polygonal numbers, Res. number theory (2018) 4: 4. https://doi.org/10.1007/s40993-018-0098-x Updated: January 19, 2021.
- (with A. Feaver, J. Liu, G. Nebe) On Kneser-Hecke operators for codes over finite chain rings, Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop, Association for Women in Mathematics Series, Springer-Verlag,
- A characterization of almost universal ternary inhomogeneous quadratic polynomials with conductor 2, J. Number Theory, 156 (2015), 247-262.
- A characterization of almost universal ternary quadratic polynomials with odd
prime power conductor, J. Number Theory, 141 (2014), 202-213.
- (with W. K. Chan) Almost universal ternary sums of squares and triangular numbers, Quadratic and Higher Degree Forms, Developments in Mathematics, Springer-Verlag, (2013).
- (with K. Doerksen) Primitive prime divisors in zero orbits of polynomials, INTEGERS: The Online Journal of Combinatorial Number Theory, 12 (2012)