Research

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