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 Faseeh Ahmad, Jennifer Cain, Gregory Carroll, and Lily Fang), The arithmetic basilica: a quadratic PCF arboreal Galois group, (submitted). Available at ArXiv:1909.00039Teaching and Research Experience:
 (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 (N.S.) 56, #4 (2019), 611-685. Also available at ArXiv:1806.04980
 (with John J. Benedetto), Frames of translates of number-theoretic groups, Journal of Geometric Analysis, 2019. Online First: DOI 10.1007/s12220-019-00234-y . Also 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, 2015-2019.Recent and Upcoming Invited Talks:
-- NSF Grant, Amherst College, 2012-2015.
-- NSF Grant, Amherst College, 2009-2012.
-- Arithmetic Dynamics International Online Seminar (ADIOS), July 15, 2020.
Brown University Mathematics Colloquium, April 1, 2020.(Cancelled due to COVID19.)
-- Oregon Number Theory Days (2 plenary talks), Oregon State University, February 29, 2020.
-- Mathematics Colloquium, Linnaeus University, Växjö, Sweden, February 5, 2020.
-- AMS Special Session on Arithmetic Dynamics at the AMS-MAA Joint Meetings in Denver, January 17, 2020.
-- Five College Number Theory Seminar (Amherst, MA), November 12, 2019.
-- Amherst College Mathematics and Statistics Colloquium, October 10, 2019.
-- Special Session on Arithmetic Dynamics at the AMS Spring Central and Western Joint Sectional Meeting at the University of Hawaii at Manoa, March 22, 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.