All of my research involves number theory of various flavors. I am very interested in how sub-rings of quaternion algebras over number fields can be used to construct groups, manifolds, and sphere packings. I also have interest in additive number theory and logic.

## Papers:

- Generating Hyperbolic Isometry Groups by Elementary Matrices. Communications in Algebra, (2023)
- Asymptotic Density of Apollonian-Type Packings (co-authored with M. Litman). arXiv: 2108.06358
- The Ulam Sequence of Linear Integer Polynomials. Journal of Integer Sequences, 24 (2021).
- Algebraic Groups Constructed from Rings with Involution. arXiv:2005.00679
- The Twisted Euclidean Algorithm: Applications to Number Theory and Geometry. Journal of Algebra, 569 (2021) 823-855.
- Quaternion Orders and Sphere Packings. Journal of Number Theory, 204 (2019) 41-98.
- Rigidity of Ulam Sets and Sequences (co-authored with J. Hinman, B. Kuca, and A. Schlesinger). Involve, 12(3) (2019) 521-539.
- The Unreasonable Rigidity of Ulam Sequences (co-authored with J. Hinman, B. Kuca, and A. Schlesinger). Journal of Number Theory 194 (2019) 409-425.
- Orders of Quaternion Algebras with Involution. Journal of Number Theory, 183 (2018) 249-268.
- A Corrigendum to Unreasonable Slightness. The American Mathematical Monthly, 123(5) (2016) 482-85.

## Talks:

- New Trends in Quaternions and Octonions (virtual).
*Is Unreasonable Slightness a General Phenomenon?*. Nov. 2021. - Rutgers University (virtual).
*Is Unreasonable Slightness a General Phenomenon?*. Nov. 2021. Slides available here. - CANT 2020 (virtual).
*A Twisted Euclidean Algorithm*. May. 2020. - UC Davis (virtual).
*Algebraic Invariants of Hyperbolic 4-Orbifolds*. May. 2020. - Graduate Center.
*The Quaternion-Bianchi Dictionary.*Nov. 2019. - Illustrating Number Theory and Algebra (at ICERM).
*Euclidean Algorithm in Circle/Sphere Packings*. Oct. 2019. Slides available here. Video recording available here. - Journées sur les Arithmétiques Faibles (at CUNY).
*Applications of Model Theory to Families of Integer Sequences*. May. 2019 - Graduate Center.
*Hyperbolic Isometry Groups and Quaternion Algebras.*Apr. 2019. - Graduate Center.
*(Searching for)**Applications of Orders with Involution.*Mar. 2019. Slides available here. - Bronx Community College.
*Two Trivial Problems and an Impossible One*. Dec. 2018. Slides available here. - Integers 2018.
*Rigidity in the Ulam Sequence*. Oct. 2018. Slides available here. - New York Logic Model Seminar at CUNY.
*Rigidity in the Ulam Sequence.*Sep. 2018. - CANT at CUNY.
*Rigidity in the Ulam Sequence*. May 2018. - Rutgers University.
*Connections between Integral Sphere Packings and Quaternion Algebras.*May 2018. Slides available here. - Dartmouth College.
*Quaternion Algebras and Integral Sphere Packings*. Feb. 2017. - University of Colorado, Boulder.
*Quaternion Algebras and Integral Sphere Packings*. Oct. 2016.

## REU Students:

- 2017 SUMRY (
*The Unreasonable Rigidity of Ulam Sets)*:- Joshua Hinman
- Borys Kuca
- Alexander Schlesinger

- 2016 SUMRY (
*Embeddings of Imaginary Quadratic Fields into Rational Quaternion Algebras*):- Alexander Schlesinger
- Rose Mintzer-Sweeney
- Katherine Xiu

- 2015 DIMACS REU (
*Cymatics-Chlandi plates*):- Lolly Kenigsberg