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.
- The Twisted Euclidean Algorithm: Applications to Number Theory and Geometry. arXiv: 2001.06061
- The Ulam Sequence of the Integer Polynomial Ring. arXiv: 1910.00109
- Algebraic Groups Constructed from Rings with Involution. arXiv:2005.00679
- 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.
- 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.
- 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