Ph.D., May 1998

Brown University, Mathematics

Dissertation: Fatou Components inp-adic dynamics

Advisor: Joseph Silverman

Sc.M., May 1996

Brown University, Mathematics

A.B., June 1993

Harvard University

Major: Mathematics

Magna Cum Laude, Phi Beta Kappa

Dynamics in One Non-Archimedean Variable, Graduate Studies in Mathematics, Volume 198, American Mathematical Society, Providence, 2019.

[38] (with Su-Ion Ih), A finiteness propoerty of postcritically finite unicritical polynomials, (submitted). Available at ArXiv:2010.15941

[37] (with Su-Ion Ih), Discreteness of postcritically finite maps inp-adic moduli space, (submitted). Available at ArXiv:2005.04656

[36] (with Faseeh Ahmad, Jennifer Cain, Gregory Carroll, and Lily Fang), The arithmetic basilica: a quadratic PCF arboreal Galois group,Journal of Number Theory, accepted pending revisions. Available at ArXiv:1909.00039

[35] (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

[34] Dynamics in One Non-Archimedean Variable, Graduate Studies in Mathematics, Volume 198, American Mathematical Society, Providence, 2019.

[33] (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

[32] (with Jamie Juul), Odoni's conjecture for number fields,Bulletin of the London Mathematical Society51, #2 (2019), 237-250. Published online 2018; DOI: https://doi.org/10.1112/blms.12225 Also available at ArXiv:1803.01987

[31] (with Xander Faber, Benjamin Hutz, Jamie Juul, and Yu Yasufuku), A large arboreal Galois representation for a cubic postcritically finite polynomial,Research in Number Theory3, #29 (2017). Open access: https://doi.org/10.1007/s40993-017-0092-8

[30] (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 Systems37(2017), 59-78.

[29] Attaining potentially good reduction in arithmetic dynamics,International Mathematics Research Notices2015, #22, 11828-11846.

[28] (with Ruqian Chen, Trevor Hyde, Yordanka Kovacheva, and Colin White), Small dynamical heights for quadratic polynomials and rational functions,Experimental Mathematics23(2014), 433-447.

[27] A criterion for potentially good reduction in non-archimedean dynamics,Acta Arithmetica165(2014), 251-256.

[26] (with Patrick Ingram, Rafe Jones, and Alon Levy), Attracting cycles inp-adic dynamics and height bounds for postcritically finite maps,Duke Mathematical Journal163(2014), 2325-2356.

[25] (with Dragos Ghioca, Pär Kurlberg, Ben Hutz, Thomas Scanlon, and Tom Tucker), Periods of rational maps modulo primes,Mathematische Annalen355(2013), 637-660.

[24] (with John Benedetto and Joseph Woodworth), Optimal ambiguity functions and Weil's exponential sum bound,Journal of Fourier Analysis and Applications18(2012), 471-487.

[23] (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.

[22] 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 .

[21] (with Dragos Ghioca, Pär Kurlberg, and Tom Tucker), A gap principle for dynamics,Compositio Mathematica146(2010), 1056-1072.

[20] (with Ben Dickman, Sasha Joseph, Ben Krause, Dan Rubin, and Xinwen Zhou), Computing points of small height for cubic polynomials,Involve2(2009), 37-64.

[19] Review:The Arithmetic of Dynamical Systems, by Joseph H. Silverman,Bulletin of the American Mathematical Society46(2009), 157-164.

[18] (with Dragos Ghioca, Pär Kurlberg, and Tom Tucker), A Case of the Dynamical Mordell-Lang Conjecture, with an appendix by Umberto Zannier;Mathematische Annalen352(2012), 1-26.

[17] An Ahlfors Islands Theorem for non-archimedean meromorphic functions,Transactions of the American Mathematical Society360(2008), 4099-4124.

[16] (with Jean-Yves Briend and Hervé Perdry), Dynamique des polynômes quadratiques sur les corps locaux,Journal de Théorie des Nombres de Bordeaux19(2007), 325-336.

[15] Preperiodic points of polynomials over global fields,Journal für die Reine und Angewandte Mathematik,608(2007), 123-153.

[14] (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)

[13] Wandering domains in non-archimedean polynomial dynamics,Bulletin of the London Mathematical Society38(2006), 937-950.

[12] Heights and preperiodic points of polynomials over function fields,International Mathematics Research Notices2005, #62, 3855-3866.

[11] Wandering domains and nontrivial reduction in non-archimedean dynamics,Illinois Journal of Mathematics49(2005), 167-193.

[10] (with John Benedetto ), A wavelet theory for local fields and related groups,The Journal of Geometric Analysis14(2004), 423-456.

[9] Examples of wavelets for local fields, in C. Heil, P. Jorgensen, D. Larson, eds.,Wavelets, Frames, and Operator Theory, Contemporary Mathematics345, AMS, Providence, 2004, pages 27-47.

[8] Non-archimedean holomorphic maps and the Ahlfors Islands Theorem,American Journal of Mathematics125(2003), 581-622.

[7] Examples of wandering domains inp-adic polynomial dynamics,Comptes Rendus Mathématique. Académie des Sciences. Paris,335(2002), 615-620.

[6] Components and periodic points in non-archimedean dynamics,Proceedings of the London Mathematical Society(3),84(2002), 231-256.

[5] An elementary product identity in polynomial dynamics,The American Mathematical Monthly108(2001), 860-864.

[4] Reduction, dynamics, and Julia sets of rational functions,Journal of Number Theory86(2001), 175-195.

[3] Hyperbolic maps inp-adic dynamics,Ergodic Theory and Dynamical Systems,21(2001), 1-11.

[2]p-adic dynamics and Sullivan's No Wandering Domains theorem,Compositio Mathematica122:3 (2000), 281-298.

[1] (with William Goldman), The topology of the relative character varieties of a quadruply-punctured sphere,Experimental Mathematics8:1 (1999), 85-103.

Amherst College, 2014-present, Professor.

Amherst College, 2008-2014, Associate Professor.

Amherst College, 2001-2008, Assistant Professor.

Boston University, 2000-2002. NSF Mathematical Sciences Research Postdoc, to studyp-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.

-- NSF Grant, Amherst College, 2012-2015.

-- NSF Grant, Amherst College, 2009-2012.

-- AMS Special Session on Current Trends in Arithmetic Dynamics at the AMS-MAA Joint Meetings, Online, January 9, 2021.

-- Amherst College Mathematics Colloquium, Online, November 12, 2020.

-- 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 17~~February 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.