University of California, Davis, Department of Mathematics
Currently, my work focuses on the interplay between geometric group theory and number theory:
I study arithmetic properties of integer matrix groups such as various monodromy groups and hyperbolic reflection groups. This usually involves combining arithmetic tools with the geometry inherent to these groups. More recently these interests have led me to investigate how generic certain properties of such groups are.
See this article in the American Scientist, or, if you speak French, this interview in the French science magazine La Recherche for an overview of the kinds of problems I considered in my thesis.Publications and Preprints:
- Local-global principles in circle packings
(with K. Stange and X. Zhang), to appear in Compositio Mathematica - The dynamics of Super-Apollonian continued fractions
(with S. Chaubey, R. Hines, and K. Stange)
To appear in Transactions of the American Mathematical Society - 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Unpublished Notes: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