Rob Benedetto

Ph.D., May 1998
Brown University, Mathematics
Dissertation: Fatou Components in p-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. (Also appears as item [34] below.)
Publications (click here to see abstracts or pdf files):
[45] (with William DeGroot, Xinyu Ni, Jesse Seid, Annie Wei, and Samantha Winton, Arboreal Galois groups for cubic polynomials with colliding critical points, (submitted). Available at ArXiv:2404.04034
[44] (with Dragos Ghioca, Jamie Juul), and Tom Tucker), Specializations of iterated Galois groups of PCF rational functions, (submitted). Available at ArXiv:2309.00840
[43] (with Anna Dietrich), Arboreal Galois groups for quadratic rational functions with colliding critical points, Mathematische Zeitschrift, accepted. Available at ArXiv:2307.16284
[42] (with Vefa Goksel), Misiurewicz polynomials and dynamical units, Part II, Research in Number Theory, 10 article 58 (2024). Available online (open access, published June 2024) at this Springer Link. DOI: 10.1007/s40993-024-00539-0. Also available at ArXiv:2203.14431
[41] (with Vefa Goksel), Misiurewicz polynomials and dynamical units, Part I, International Journal of Number theory 19, #6 (2023), 1249-1267. DOI: 10.1142/S1793042123500616 . Also available at ArXiv:2201.07868
[40] A survey of non-archimedean dynamics, Notices of the American Mathematical Society 69, #5 (2022), 715-723. Published online May 2022; DOI: 10.1090/noti2472.
[39] (with Junghun Lee), J-Stability in non-archimedean dynamics, Advances in Mathematics 397 (2022), Paper No. 108204, 33 pages. Available at ArXiv:2102.05841
[38] (with Su-Ion Ih), A finiteness property of postcritically finite unicritical polynomials, Mathematical Research Letters 30, #2 (2023), 295-317. Published online 2023; DOI: Also available at ArXiv:2010.15941
[37] (with Su-Ion Ih), Discreteness of postcritically finite maps in p-adic moduli space, Transactions of the American Mathematical Society 377, #3 (2024), 2027-2048. Also 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 238 (2022), 842--868. Published online 2021; DOI: Also 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 30, #4 (2020), 4126-4149. Online First, 2019: 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 Society 51, #2 (2019), 237-250. Published online 2018; DOI: 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 Theory 3, #29 (2017). Open access:
[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 Systems 37 (2017), 59-78.
[29] Attaining potentially good reduction in arithmetic dynamics, International Mathematics Research Notices 2015, #22, 11828-11846.
[28] (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.
[27] A criterion for potentially good reduction in non-archimedean dynamics, Acta Arithmetica 165 (2014), 251-256.
[26] (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.
[25] (with Dragos Ghioca, Pär Kurlberg, Ben Hutz, Thomas Scanlon, and Tom Tucker), Periods of rational maps modulo primes, Mathematische Annalen 355 (2013), 637-660.
[24] (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.
[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 .
[21] (with Dragos Ghioca, Pär Kurlberg, and Tom Tucker), A gap principle for dynamics, Compositio Mathematica 146 (2010), 1056-1072.
[20] (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.
[19] Review: The Arithmetic of Dynamical Systems, by Joseph H. Silverman, Bulletin of the American Mathematical Society 46 (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 Annalen 352 (2012), 1-26.
[17] An Ahlfors Islands Theorem for non-archimedean meromorphic functions, Transactions of the American Mathematical Society 360 (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 Bordeaux 19 (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 Society 38 (2006), 937-950.
[12] Heights and preperiodic points of polynomials over function fields, International Mathematics Research Notices 2005, #62, 3855-3866.
[11] Wandering domains and nontrivial reduction in non-archimedean dynamics, Illinois Journal of Mathematics 49 (2005), 167-193.
[10] (with John Benedetto ), A wavelet theory for local fields and related groups, The Journal of Geometric Analysis 14 (2004), 423-456.
[9] 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.
[8] Non-archimedean holomorphic maps and the Ahlfors Islands Theorem, American Journal of Mathematics 125 (2003), 581-622.
[7] Examples of wandering domains in p-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 Monthly 108 (2001), 860-864.
[4] Reduction, dynamics, and Julia sets of rational functions, Journal of Number Theory 86 (2001), 175-195.
[3] Hyperbolic maps in p-adic dynamics, Ergodic Theory and Dynamical Systems, 21 (2001), 1-11.
[2] p-adic dynamics and Sullivan's No Wandering Domains theorem, Compositio Mathematica 122:3 (2000), 281-298.
[1] (with William Goldman), The topology of the relative character varieties of a quadruply-punctured sphere, Experimental Mathematics 8:1 (1999), 85-103.
Teaching and Research Experience:
Amherst College, 2020-present, William J. Walker Professor of Mathematics
Amherst College, 2014-2020, Professor.
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.
Recent Awards and Honors:
-- NSF Grant, Amherst College, 2021-2024.
-- NSF Grant, Amherst College, 2015-2019.
-- NSF Grant, Amherst College, 2012-2015.
-- NSF Grant, Amherst College, 2009-2012.
Recent and Upcoming Invited Talks:
-- Dynamical Days in Montreal Workshop, Concordia University, June 5, 2024.
-- Towson University Algebra Seminar, April 25, 2024.
-- Towson University Math Colloquium, April 24, 2024.
-- Front Range Number Theory Day, University of Colorado, Boulder, April 13, 2024.
-- Front Range Algebraic Geometry and Number Theory Seminar, Colorado State University, April 11, 2024.
-- Oklahoma State University Colloquium, March 8, 2024.
-- Oklahoma State University Number Theory Seminar, March 7, 2024.
-- Brown University Algebra Seminar, Feburary 12, 2024.
-- University of Maryland Mathematics Colloquium, February 7, 2024.
-- Amherst College Mathematics Colloquium, September 20, 2023.
-- Harvard University Algebraic Dynamics Seminar, March 30, 2023.
-- AMS Special Session on Recent Advances in Arithmetic Dynamics at the Joint Mathematics Meetings, Boston, January 6, 2023.
-- Oklahoma State University Number Theory Seminar, Online, March 24, 2022.
-- Harvard University Algebraic Dynamics Seminar, Online, March 19, 2021.
-- 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.