Mathematics Research

Mathematical Interests

I work with arithmetic groups and quadratic forms, a fascinating area of mathematics that lies in the intersection of differential geometry and number theory. As such, my interests span a wide collection of areas including spectral geometry, geometric group theory, Lie theory, algebraic number theory, and class field theory. My thesis advisor was Matthew Stover.

Preprints and Works In Progress

  1. Arithmeticity and Hidden Symmetries of Fully Augmented Pretzel Link Complements (joint w. C. Millichap and R. Trapp)
    1. (2018) arXiv Preprint
  2. Systole inequalities up congruence towers for arithmetic locally symmetric spaces (joint w. S. Lapan and B. Linowitz)
    1. (2017) arXiv Preprint
  3. Counting Commensurability Classes of Codimension One Subspaces (joint w. B. Linowitz)
    1. (in preparation)
  4. Totally Geodesic Spectra of Quaternionic Hyperbolic Orbifolds
    1. (2015) arXiv Preprint


  1. Constructing Geometrically Equivalent Hyperbolic Orbifolds (joint w. D.B. McReynolds and M. Stover)
    1. (2017) Algebraic & Geometric Topology, Volume 17, no. 2, 831-846.
    2. (2015) arXiv Preprint
  2. On the isospectral orbifold-manifold problem for nonpositively curved locally symmetric spaces (joint w. B. Linowitz)
    1. (2017) Geometriae Dedicata, Volume 188, no. 1.
    2. (2015) arXiv Preprint
  3. Systolic Surfaces of Arithmetic Hyperbolic 3-Manifolds (joint w. B. Linowitz)
    1. (2017) In the tradition of Ahlfors-Bers. VII, 215–224, Contemp. Math., 696, Amer. Math. Soc., Providence, RI.
    2. (2015) arXiv Preprint
  4. The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces (joint w. B. Linowitz and P. Pollack)
    1. (2015) New York Journal of Math, Volume 21, 955-972.
    2. (2015) arXiv Preprint
  5. Totally Geodesic Spectra of Arithmetic Hyperbolic Spaces
    1. (2017) Transactions of the American Mathematical Society, 369, pp. 7549-7588
    2. (2014) arXiv Preprint
  6. Division Algebras With Infinite Genus
    1. (2014) Bulletin of the London Mathematical Society, Volume 46, no. 3, 463-468.
    2. (2013) arXiv Preprint

Unpublished Manuscripts

  1. On The Totally Geodesic Commensurability Spectrum of Arithmetic Locally Symmetric Spaces
    1. (2013) PhD Thesis. University of Michigan.