University of California, Davis, Department of Mathematics
My research is currently supported by NSF Grant DMS-2154624.
I study arithmetic properties of thin groups. These groups lurk in the background of problems on Apollonian circle packings, Markoff triples, and more Recently, some of the difficult combinatorial problems related to such groups have led me to venture into cryptography as well.
Current and past graduate students:- Matt Litman (2019-2023)
- Colby Brown (2022-)
Current and past postdocs:
- Xin Zhang (2015-2018)
- Daniel Martin (2020-2023)
- Connectivity of Markoff mod-p graphs and maximal divisors (with J. Eddy, M. Litman, D. Martin, N. Tripeny), submitted.
- Orbits on K3 Surfaces of Markoff Type (with M. Litman, J. Silverman, A. Tran), Experimental Math. DOI
- A Cryptographic Hash Function based on Markoff Triples (with K. Lauter, M. Litman, A. Tran), Mathematical Cryptology (2021)
- Local-global principles in circle packings
(with K. Stange and X. Zhang), Compositio Mathematica (2019) - The dynamics of Super-Apollonian continued fractions
(with S. Chaubey, R. Hines, and K. Stange)
Transactions of the American Mathematical Society (2019) - Generic thinness in finitely generated subgroups of SL_n(Z)
(with I. Rivin)
International Mathematics Research Notices doi: 10.1093/imrn/rnw136 (2016) - Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions (with C. Meiri and P. Sarnak)
Journal of the European Mathematical Society, 8 (2014), 1617-1671 - The ubiquity of thin groups
in: Proceedings of MSRI conference on Thin Groups and Super-strong Approximation - Counting problems in Apollonian packings
Bulletin of the American Mathematical Society, 50 (2013), 229-266 - On representation of integers by binary quadratic forms
(with J. Bourgain)
International Mathematics Research Notices (2012), 5505-5553 - Some experiments with integral Apollonian circle packings
(with K. Sanden)
Experimental Mathematics, 20 (2011), 380-399 - A proof of the positive density conjecture for integer Apollonian circle packings
(with J. Bourgain)
Journal of the American Mathematical Society, 24 (2011), 945-967 - Strong Approximation in the Apollonian Group
Journal of Number Theory, 131 (2011), 2282-2302 - Arithmetic Properties of Apollonian Circle Packings
Ph.D. Thesis (2010), Princeton University - Longest induced cycles in circulant graphs
Electronic Journal of Combinatorics 14, no.1, Research Paper 52 (2005)
Unpublished Notes:
- A note on the density of curvatures in integer Apollonian circle packings , 2009
- Explicit formulas for the modular equation (with P. Baginski), 2004