# 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 systolic geometry, spectral geometry, geometric group theory, Lie theory, algebraic number theory, and class field theory. My thesis advisor was Matthew Stover.

Using arithmetic lattices, I study the topology and geometry of arithmetic locally symmetric spaces. Most of my research is aimed at analyzing questions from systolic and spectral geometry concerning arithmetic locally symmetric spaces. Furthermore, over the past several years, I have become increasingly interested in using computers perform computations, run searches, and create visualizations within hyperbolic geometry, in particular using the Python-based Sage.

Above is the Sage image I created of a systole curve in the principal arithmetic surface associated to the Hilbert symbol (2,3 / Q).

Jeff with some Zipper-gons

**Publications**

Universal systole bounds for arithmetic locally symmetric spaces (joint w. S. Lapan and B. Linowitz)

Systole inequalities up congruence towers for arithmetic locally symmetric spaces (joint w. S. Lapan and B. Linowitz)

(2022) To appear in Communication in Analysis and Geometry.

Arithmeticity and Hidden Symmetries of Fully Augmented Pretzel Link Complements (joint w. C. Millichap and R. Trapp)

Constructing Geometrically Equivalent Hyperbolic Orbifolds (joint w. D.B. McReynolds and M. Stover)

On the isospectral orbifold-manifold problem for nonpositively curved locally symmetric spaces (joint w. B. Linowitz)

Systolic Surfaces of Arithmetic Hyperbolic 3-Manifolds (joint w. B. Linowitz)

The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces (joint w. B. Linowitz and P. Pollack)

Totally Geodesic Spectra of Arithmetic Hyperbolic Spaces

Division Algebras With Infinite Genus

**Unpublished Manuscripts**

On The Totally Geodesic Commensurability Spectrum of Arithmetic Locally Symmetric Spaces

**Preprints and Works In Progress**

*Computing Cheeger Constants of Principal Congruence Surfaces (joint w. B. Benson and G. Lakeland)**(in preparation)*

Totally Geodesic Spectra of Quaternionic Hyperbolic Orbifolds