Research

My primary interest is in studying arithmetic groups and their connections to geometry and model theory; however, I do have some projects in additive number theory and model theory. Many of the projects that I have pursued required extensive computations, which has led me to study computational and numerical methods as well.

Papers:

• Distribution of Ulam Words up to Length 30. Integers, (2025)
• 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 with Available Slides:

• Joint Math Meetings. Ulam Sequences: Chaos and Order and Connections Between the Two. Jan. 2024. Slides available here.
• Rutgers University (virtual). Is Unreasonable Slightness a General Phenomenon?. Nov. 2021. Slides available here.
• Illustrating Number Theory and Algebra (at ICERM). Euclidean Algorithm in Circle/Sphere Packings. Oct. 2019. Slides available here. Video recording is available here.
• 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.
• Rutgers University. Connections between Integral Sphere Packings and Quaternion Algebras. May 2018. Slides available here.

REU (and REU-like) Students:

• 2024 Bates College (Structures in Ulam-like Sequences):
  • Paul Adutwum
  • Hopper Clark
  • Ro Emerson
  • Axelle Tougouma
• 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