Ph.D., May 1998Book:
Brown University, Mathematics
Dissertation: Fatou Components in p-adic dynamics
Advisor: Joseph Silverman
Sc.M., May 1996
Brown University, Mathematics
A.B., June 1993
Magna Cum Laude, Phi Beta Kappa
Dynamics in One Non-Archimedean Variable, Graduate Studies in Mathematics, Volume 198, American Mathematical Society, Providence, 2019.Publications (click here to see abstracts or pdf files):
 (with Patrick Ingram, Rafe Jones, Michelle Manes, Joseph Silverman, and Tom Tucker), Current trends and open problems in arithmetic dynamics, Bulletin of the American Mathematical Society, to appear. Available at ArXiv:1806.04980Teaching and Research Experience:
 (with John J. Benedetto), Frames of translates of number-theoretic groups, submitted. Available at ArXiv:1804.07783
 (with Jamie Juul), Odoni's conjecture for number fields, Bulletin of the London Mathematical Society 51, #2 (2019), 237-250. Published online 2018; DOI: https://doi.org/10.1112/blms.12225 Also available at ArXiv:1803.01987
 (with Xander Faber, Benjamin Hutz, Jamie Juul, and Yu Yasufuku), A large arboreal Galois representation for a cubic postcritically finite polynomial, Research in Number Theory 3, #29 (2017). Open access: https://doi.org/10.1007/s40993-017-0092-8
 (with Dvij Bajpai, Ruqian Chen, Edward Kim, Owen Marschall, Darius Onul, and Yang Xiao), Non-archimedean connected Julia sets with branching, Ergodic Theory and Dynamical Systems 37 (2017), 59-78.
 Attaining potentially good reduction in arithmetic dynamics, International Mathematics Research Notices 2015, #22, 11828-11846.
 (with Ruqian Chen, Trevor Hyde, Yordanka Kovacheva, and Colin White), Small dynamical heights for quadratic polynomials and rational functions, Experimental Mathematics 23 (2014), 433-447.
 A criterion for potentially good reduction in non-archimedean dynamics, Acta Arithmetica 165 (2014), 251-256.
 (with Patrick Ingram, Rafe Jones, and Alon Levy), Attracting cycles in p-adic dynamics and height bounds for postcritically finite maps, Duke Mathematical Journal 163 (2014), 2325-2356.
 (with Dragos Ghioca, Pär Kurlberg, Ben Hutz, Thomas Scanlon, and Tom Tucker), Periods of rational maps modulo primes, Mathematische Annalen 355 (2013), 637-660.
 (with John Benedetto and Joseph Woodworth), Optimal ambiguity functions and Weil's exponential sum bound, Journal of Fourier Analysis and Applications 18 (2012), 471-487.
 (with John Benedetto), The construction of wavelet sets, in Jonathan Cohen, Ahmed I. Zayed, eds., Wavelets and Multiscale Analysis: Theory and Applications, Springer, New York, 2011; pages 17-56.
 Non-archimedean dynamics in dimension one: Lecture notes, from my lectures at the 2010 Arizona Winter School. Available at http://swc.math.arizona.edu/aws/2010 .
 (with Dragos Ghioca, Pär Kurlberg, and Tom Tucker), A gap principle for dynamics, Compositio Mathematica 146 (2010), 1056-1072.
 (with Ben Dickman, Sasha Joseph, Ben Krause, Dan Rubin, and Xinwen Zhou), Computing points of small height for cubic polynomials, Involve 2 (2009), 37-64.
 Review: The Arithmetic of Dynamical Systems, by Joseph H. Silverman, Bulletin of the American Mathematical Society 46 (2009), 157-164.
 (with Dragos Ghioca, Pär Kurlberg, and Tom Tucker), A Case of the Dynamical Mordell-Lang Conjecture, with an appendix by Umberto Zannier; Mathematische Annalen 352 (2012), 1-26.
 An Ahlfors Islands Theorem for non-archimedean meromorphic functions, Transactions of the American Mathematical Society 360 (2008), 4099-4124.
 (with Jean-Yves Briend and Hervé Perdry), Dynamique des polynômes quadratiques sur les corps locaux, Journal de Théorie des Nombres de Bordeaux 19 (2007), 325-336.
 Preperiodic points of polynomials over global fields, Journal für die Reine und Angewandte Mathematik, 608 (2007), 123-153.
 (with Liang-Chung Hsia , A quotient of elliptic curves - weak Néron models for Lattès maps, in K. Komatsu and K. Hashimoto, eds., Proceedings of the 2007 Waseda Number Theory Symposium, Institute of Mathematics, Waseda University, 2007. (not peer-reviewed)
 Wandering domains in non-archimedean polynomial dynamics, Bulletin of the London Mathematical Society 38 (2006), 937-950.
 Heights and preperiodic points of polynomials over function fields, International Mathematics Research Notices 2005, #62, 3855-3866.
 Wandering domains and nontrivial reduction in non-archimedean dynamics, Illinois Journal of Mathematics 49 (2005), 167-193.
 (with John Benedetto ), A wavelet theory for local fields and related groups, The Journal of Geometric Analysis 14 (2004), 423-456.
 Examples of wavelets for local fields, in C. Heil, P. Jorgensen, D. Larson, eds., Wavelets, Frames, and Operator Theory, Contemporary Mathematics 345, AMS, Providence, 2004, pages 27-47.
 Non-archimedean holomorphic maps and the Ahlfors Islands Theorem, American Journal of Mathematics 125 (2003), 581-622.
 Examples of wandering domains in p-adic polynomial dynamics, Comptes Rendus Mathématique. Académie des Sciences. Paris, 335 (2002), 615-620.
 Components and periodic points in non-archimedean dynamics, Proceedings of the London Mathematical Society (3), 84 (2002), 231-256.
 An elementary product identity in polynomial dynamics, The American Mathematical Monthly 108 (2001), 860-864.
 Reduction, dynamics, and Julia sets of rational functions, Journal of Number Theory 86 (2001), 175-195.
 Hyperbolic maps in p-adic dynamics, Ergodic Theory and Dynamical Systems, 21 (2001), 1-11.
 p-adic dynamics and Sullivan's No Wandering Domains theorem, Compositio Mathematica 122:3 (2000), 281-298.
 (with William Goldman), The topology of the relative character varieties of a quadruply-punctured sphere, Experimental Mathematics 8:1 (1999), 85-103.
Amherst College, 2014-present, Professor.Recent Awards and Honors:
Amherst College, 2008-2014, Associate Professor.
Amherst College, 2001-2008, Assistant Professor.
Boston University, 2000-2002. NSF Mathematical Sciences Research Postdoc, to study p-adic and number-theoretic dynamics, under the mentorship of David Rohrlich and Bob Devaney.
University of Rochester, 1998-2000. Visiting Assistant Professor.
Brown University, 1993-1998. Graduate study.
University of Maryland, Summers of 1992, 1993. Undergraduate research under the supervision of William Goldman.
-- NSF Grant, Amherst College, 2015-2018.Recent and Upcoming Invited Talks:
-- NSF Grant, Amherst College, 2012-2015.
-- NSF Grant, Amherst College, 2009-2012.
-- Special Session on Arithmetic Dynamics at the AMS Spring Central and Western Joint Sectional Meeting at the University of Hawaii at Manoa, March 23, 2019.
-- Workshop on Complex and Arithmetic Dynamics, Northwestern University, May 18, 2018.
-- Colby College Mathematics Colloquium, April 2, 2018.
-- Brown University Algebra Seminar, March 12, 2018.
-- University of Connecticut Algebra Seminar,
January 17February 28, 2018. (Postponed due to weather.)
-- AMS Special Session on A Showcase of Number Theory at Liberal Arts Colleges at the AMS-MAA Joint Meetings in San Diego, January 11, 2018.
-- University of Colorado, Boulder, Number Theory Seminar, October 10, 2017.
-- Special Session on Arithmetic Dynamics at the Mathematical Congress of the Americas, Montreal, July 26, 2017.
-- Mini Workshop on Arithmetic Dynamics, University of Michigan, May 16, 2017.
-- University of Maine, Orono, Mathematics Colloquium, March 29, 2017.
-- Five College Number Theory Seminar, Amherst, MA, Feb 14, 2017.
-- Workshop on Interactions between Model Theory and Arithmetic Dynamics, Fields Institute, Toronto, July 29, 2016.
-- AMS Special Session on Arithmetic Dynamics at the AMS-MAA Joint Meetings in Seattle, January 6, 2016.
-- Workshop on Arithmetic Dynamics, University of Michigan, December 6, 2015.
-- Arithmetic 2015: Silvermania Conference, Brown University, August 11, 2015.
-- SURF (Summer Undergraduate Research Fellowships) seminar at Amherst College, June 26, 2015.
-- Special Session on Number Theory in Ergodic Theory and Dynamical Systems at the AMS Spring Eastern Sectional Meeting at Georgetown University, March 7, 2015.
-- Amherst College Mathematics and Statistics Colloquium, November 10, 2014.
-- Binghamton University Arithmetic Seminar, October 10, 2014.
-- MathPath at Mount Holyoke College (plenary speaker), July 15, 2014.
-- Mount Holyoke College REU Seminar, June 18, 2014.
-- Elliptic Curves @ UConn (Instructional Conference), at the University of Connecticut, May 17, 2014.