Rafe Jones

Carleton College
Department of Mathematics
1 North College St.
Northfield, MN 50557

rfjones@carleton.edu
http://www.people.carleton.edu/~rfjones/(507) 222 - 4450

Employment
CARLETON COLLEGE, Northfield, MN
A ssistant Professor, 2012 – 2016
A ssociate Professor, 2016 – present
COLLEGE OF THE HOLY CROSS, Worcester, MAA ssistant Professor, 2008 – 2012
UNIVERSITY OF WISCONSIN, Madison, WI
V IGRE V an V leck V isiting A ssistant Professor, 2005 – 2008

Education
BROWN UNIVERSITY, Providence, RI
Ph.D., Mathematics, May 2005
Advisor: Joseph Silverman
Thesis Title: Galois Martingales and the Density of the p-adic Hyperbolic Mandelbrot Set
ÉCOLE NORMALE SUPÉRIEURE, Paris, France
Visiting student, Département de Mathématiques et Applications, 1998-99
AMHERST COLLEGE, Amherst, MAB. A ., Mathematics and French, June 1998
Summa Cum Laude, with Departmental Distinction in Mathematics
Phi Beta Kappa

Research Interests
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Galois theory and irreducibility of polynomials (particularly involving iterated morphisms)
Discrete dynamics (particularly iteration of rational functions over global and finite fields)
Automorphism groups of rooted trees and iterated monodromy groups
Elliptic curves and torsion fields
Recurrence sequences

Refereed Publications
1. Rafe Jones, Jordan Cahn, and Jacob Spear. Perfect powers in orbits of rational functions: cases of anarithmetic dynamical Mordell-Lang-type conjecture. Submitted.
2. Rafe Jones and Alon Levy. Eventually stable rational functions. Int. J. Number Theory 13 (9) (2017),2299-2318.
3. Benjamin Breen, Rafe Jones, Thomas Occhipinti, and Michelle Yuen. Wild ramification in a family oflow-degree extensions arising from iteration. JP J. A lgebra Number Theory A ppl. 37(1) (2015), 69-104.
4. Spencer Hamblen, Rafe Jones, and Kalyani Madhu. The density of primes in orbits of z d + c. Int. Math.
Res. Not. IMRN 7 (2015) 1924–1958.

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5. Rafe Jones. Fixed-point-free elements of iterated monodromy groups. Trans. A mer. Math. Soc. 367(3)(2015) 2023-2049.
6. Robert Benedetto, Patrick Ingram, Rafe Jones, and Alon Levy. Attracting cycles in p-adic dynamics andheight bounds for post-critically finite maps. Duke Math. J. 163(13) (2014) 2325-2356.
7. Rafe Jones and Michelle Manes. Galois theory of quadratic rational functions. Comment. Math. Helv.
89(1) (2014), 173-213.
8. Katharine Chamberlin, Emma Colbert, Sharon Frechette, Patrick Hefferman, Rafe Jones, and SarahOrchard. Newly reducible iterates in families of quadratic polynomials. Involve: A Journal of Mathematics
5(4) (2012), 481-495.
9. Rafe Jones. An iterative construction of irreducible polynomials reducible modulo all primes. J. A lgebra
369 (2012), 114-128.
10. Nigel Boston and Rafe Jones. Settled polynomials over finite fields. Proc. A mer. Math. Soc. 140(6)(2012), 1849-1863.
11. Rafe Jones. Achievement sets of sequences. A mer. Math. Monthly 118(6) (2011), 508-521.
12. Rafe Jones and Han Peters. Blocks of monodromy groups in complex dynamics. Geom. Dedicata 150(1)(2011), 137–150.
13. Rafe Jones and Jeremy Rouse (with an appendix by Jeffrey D. Achter). Galois theory of iterated
endomorphisms. Proc. Lond. Math. Soc. 100(3) (2010), 763-794.
14. Xander Faber, Benjamin Hutz, Patrick Ingram, Rafe Jones, Michelle Manes, Thomas J. Tucker, andMichael Zieve. Uniform bounds on pre-images under quadratic dynamical systems. Math. Res. Lett. 16(1)(2009), 87-101.
15. Nigel Boston and Rafe Jones. The image of an arboreal Galois representation. Pure A ppl. Math. Q. 5(1)(Special Issue: In honor of Jean-Pierre Serre, Part 2 of 2) (2009) 213—225.
16. Rafe Jones. The density of prime divisors in the arithmetic dynamics of quadratic polynomials. J. Lond.
Math. Soc. (2) 78(2) (2008), 523—544.
17. Rafe Jones. Iterated Galois towers, their associated martingales, and the p-adic Mandelbrot set.
Compositio Math. 43(5) (2007), 1108—1126.
18. Nigel Boston and Rafe Jones. Arboreal Galois representations. Geom. Dedicata 124(1) (2007), 27—35.
19. Rafe Jones and Jan Pearce. A postmodern view of fractions and the reciprocals of Fermat primes,
Mathematics Magazine 73 (2000), 83-97 (Awarded Allendoerfer Prize for excellence in mathematical
exposition by the MAA. See Fellowships and Awards.) Reprinted in Arthur Benjamin and Ezra Graham,editors, Biscuits of Number Theory, Dolciani Mathematical Expositions number 34, pages 23-38.
Mathematical Association of America, Washington, DC, 2009.

Other Publications
1. Rafe Jones. Enlightening Symbols: A Short History of Mathematical Notation – A Book Review. Math.
Intelligencer 38(2) (2016), 85-86.
2. Rafe Jones. Galois representations from pre-image trees: an arboreal survey. Publications Mathématiquesde Besançon: A lgèbre et Théorie des Nombres (2013) special issue devoted to the proceedings of theconference “Number theory and its applications,” held at the Centre International de Rencontres
Mathématiques, Luminy, France, January 2012.
3. Rafe Jones. Numbers: A V ery Short Introduction – A Book Review. Notices of the A MS January 2012:50-51 (this review furnished the inspiration for the cover of the January 2012 issue).
4. Rafe Jones. Do the math: a new way of thinking about the economy. Editorial in the W orcester Telegram& Gazette, February 10, 2009 and The Providence Journal, February 18, 2009.

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5. Rafe Jones. Galois actions on rooted trees. Appears in Pro-p extensions of global fields and pro-p groups.
Abstracts from the workshop held May 21–27, 2006. Organized by Nigel Boston, John Coates, and Fritz
Grunewald. Oberwolfach Rep. 3 (2006), no. 2, 1463—1535.
6. Rafe Jones. Ninety-one years [Poem]. New Millennium W ritings 14 (June 2004): 180-81.
7. Rafe Jones. In Code: A Mathematical Journey – A Book Review. Notices of the A MS April 2003: 460461.
8. Rafe Jones. A summer at discovery.com. Notices of the A MS June/July 2002: 694-696.
9. Rafe Jones. Stern perfection: mathematics as a fine art. The Catalyst Special Edition, Brown University,
2002.

GrantsNSF Grant DMS-0852826. Title: “Arboreal Galois representations and applications to arithmetic dynamics,”$84,353, August 20, 2008 – July 31, 2011. (Extended to July 31, 2012)
NSA Young Investigators. Title: “Arboreal Galois representations and applications to arithmetic dynamics,”
August 2008-August 2010. Grant awarded but declined due to rules against holding NSA and NSF grants
concurrently.

Fellowships and Awards
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Project NExT National Fellow, 2008-2009
Mathematics Department Outstanding Teaching Award, Brown University, December 2004
Dissertation Fellowship, Brown University, Fall 2003
VIGRE Graduate Trainee, Brown University, Spring 2002 and Spring 2003
American Association for the Advancement of Science/American Mathematical Society Mass Media
Fellowship; host site was Discovery.com in Bethesda, MD. Summer 2001.
Allendoerfer Prize for mathematical exposition. Awarded by the Mathematical Association of America
for the paper "A Postmodern View of Fractions and the Reciprocals of Fermat Primes," June 2001
University Fellowship, Brown University, Fall 1999-Spring 2000

Recent Presentations and Conferences
Portland State University Maseeh Mathematics and Statistics colloquium, October 2017, invited speaker.
Washington and Lee University colloquium, March 2017, invited speaker.
American Institute of Mathematics workshop on “The Galois theory of orbits in arithmetic dynamics” PaloAlto, CA. Leadoff speaker. May 2016.
AMS Special Session, “Arithmetic Dynamics”, Joint Mathematics Meetings, Seattle, WA. Invited speaker.
January 2016.
University of Rochester Number Theory seminar, November 2015, invited speaker.
Silvermania (A conference in honor of Joseph Silverman’s 60th birthday), August 2015, invited speaker.
Brown University Algebra Seminar, March 2015, invited speaker.
Front Range Algebraic Geometry and Number Theory seminar, Colorado State University, September 2014,invited speaker.
American Institute of Mathematics workshop on “Post-critically finite maps in complex and arithmeticdynamics,” Palo Alto, CA. Invited participant. March 2014.
St. Olaf undergraduate seminar, December 2013, invited speaker.
Laboratoire Mathématiques de Besançon, conference titled “Algebraic and explicit methods in numbertheory.” Invited to give a 60-minute talk. September 2013.

3Banff International Research Station, conference titled “The art of iterating rational functions over finitefields,” invited participant. March 2013.

Professional Service
Co-organizer (with Joseph Silverman and Michelle Manes) of American Institute of Mathematics weeklong
program “The Galois theory of orbits in arithmetic dynamics”, Palo Alto, CA, May 2016.
Allendoerfer Prize committee, Mathematical Association of America (2015-present)
Co-organizer (with Colleen Duffy) of AMS Special Session on Number Theory at AMS Central Sectional, EauClaire, WI, September 2014.
Co-organizer (with Farshid Hajir and Michael Bush) of AMS Special Session on Number Theory, Arithmetic
Topology, and Arithmetic Dynamics at AMS Eastern Sectional, Worcester, MA, April 2011.
Co-organizer (with Joseph Silverman and Michelle Manes) of AMS special session Arithmetic and Nonarchimedean Dynamics at Joint Mathematics Meetings, San Francisco, CA, January 2010.
Reviewer for NSA grant proposals, 2006-present (4 proposals).
Co-organizer of Project NEXT session "Putting together pre-tenure review materials," Mathfest, Portland, OR,2009.
Reviewer for Math Reviews, 2009-present (22 article reviews, 2 book reviews).
Reviewer for Zentralblatt Math, 2008-2010 (5 reviews).
Referee for the following publications (2007-present):• Proceedings of the American Mathematical Society (6 papers)• American Mathematical Monthly (3 papers)• Bulletin of the London Mathematical Society (3 papers)• Acta Arithmetica (2 papers)• Involve (2 papers)• Journal de Théorie des Nombres de Bordeaux (2 papers)• Journal of Number Theory (2 papers)• Canadian Journal of Mathematics (1 paper)• Canadian Mathematical Bulletin (1 paper)• Crelle’s Journal (1 paper)• Duke Mathematical Journal (1 paper)• Experimental Mathematics (1 paper)• Glasgow Mathematical Journal (1 paper)• Integers (1 paper)• LMS Journal of Computation and Mathematics (1 paper)• Mathematische Zeitschrift (1 paper)• New York Journal of Mathematics (1 paper)• Research in Number Theory (1 paper)• Topological Methods in Nonlinear Analysis (1 paper)

Research Mentoring
Research resulting from a comps project, summer 2014, “W ild ramification in a family of low-degree
extensions arising from iteration.”•In the winter and spring of 2014, I advised Hanan Abo Sakr, Trey Brademan, Ben Breen, and Michelle
Yuen on a comps project exploring the ramification properties of the prime 2 in extensions of Q generated
by the roots of the second iterate of the map f(x) = x 2 + c, where c is an integer. These properties hadn’t
been studied before, and are of interest to the research community. The students obtained a partial
classification of this behavior as part of their project. Ben and Michelle expressed an interest in continuing4to work on the problem, and in writing up the results for publication. I enlisted Tommy Occhipinti, one ofour visiting faculty members who is also a number theorist, to help out. Together we worked through thissurprisingly thorny question – just stating the main result takes nearly a full page!
Informal project advisor, University of Minnesota-Duluth REU, 2014, “On arboreal Galois representations ofrational functions.”•At the request of the REU direction, Joseph Gallian, I proposed some possible projects before the summer
began. Joe then selected this one for the student Ashvin Swaminathan, and I corresponded at length withhim about his results, his proofs, and where he could look for additional problems and research directions.
I read several drafts of his paper, which is currently submitted to the Journal of Algebra.

Summer research project, 2013: “Perfect powers in orbits of rational functions.”•I worked with Carleton rising sophomores Jordan Cahn and Jacob Spear on a project studing under what
conditions a rational function with coefficients in a number field can have an orbit containing infinitely
many mth powers, for a fixed m greater than 1. This precise question evolved over discussions with Jordanand Jacob, after we started with a somewhat different and much more limited question. The students thenpushed hard to find a solution to the most general form of the question, with impressive results. We areworking on a joint article containing these results, which will be submitted for publication.

Summer research project 2012: “Newly reducible iterates in families of quadratic polynomials.”•This summer research project involved three Holy Cross rising seniors, Emma Colbert, Patrick Hefferman,and Sarah Orchard, who continued work begun by Katherine Chamberlin in her senior thesis. Prof. Sharon
Frechette co-advised the students for a portion of the summer. The three students set out to prove that for
any fixed n greater than 1, there are only finitely many choices for b such that the polynomial f(x) = x 2 + b
has its nth iterate irreducible, but its (n+1)st iterate reducible. This result answered a natural question
arising from a 2000 paper of Danielson and Fein that appeared in the Proceedings of the American
Mathematical Society. Sharon and I worked with the students to write up their results, and the paper
appeared in the journal Involve.

Senior Thesis, 2011-2012: “Irreducibility of iterates of a second degree polynomial,” by Katherine
Chamberlin.
•The project involves studying irreducibility properties of iterates of polynomials of the form f(x) = x 2 + b,
where b is a rational number. For instance, it is not known for which values of b the third iterate of f isreducible, but the second iterate is irreducible. Completing the project involved understanding field
extensions, the norm map, theorems about rational points on curves, methods for finding rational points on
hyperelliptic curves, and perhaps computations involving Grobner bases. Kate gave a poster presentation
on her work at the 2012 Joint Mathematics Meetings.

University of Georgia V IGRE Summer School in A rithmetic Dynamics, May 2011•

As one of the plenary speakers at this summer school, I designed several projects for the graduate students
in attendance to work on, and consulted with them daily for a week to help their progress. The projects
involved areas of currently active research.

Senior Thesis, 2009-2010: “The proportion of k-cycles for polynomials modulo primes,” by Jonathan Root.
•This project that involved understanding current research literature and proving two new results, one ingroup theory and one involving the irreducibility of certain polynomials. Jonathan wrote up his work in
the form of a 15-page paper. He has also presented his results in a seminar talk at Holy Cross and anotherin the Brown University graduate student seminar. In addition to providing the idea for the project andmathematical guidance, I gave feedback on numerous drafts of both the thesis and presentation. Jonathan
received the Crompton Gold Medal for his project, which is awarded annually to a Holy Cross student forthe best scientific essay or research paper.

5A rizona W inter School, March 2010.
•Assisted on several projects having to do with Joseph Silverman’s lectures on arithmetic dynamics. Oneproject on which I worked particularly closely has led to the manuscript “Graph components and dynamics
over finite fields” by Ryan Flynn and Derek Garton, which has been submitted to the Proceedings of theA merican Mathematical Society. In addition to the work at the Winter School, I read and gave commentson several versions of the manuscript.

University of W isconsin Research Experience for Undergraduates (REU) in Number Theory, Summer 2006.
•Designed and advised students on two eight-week projects dealing with arithmetic dynamics: one on
primitive prime divisors in quadratic arithmetical dynamical systems, and one on Galois groups of iterates
of a special family of polynomials. The former project resulted in Brian Rice’s paper “Primitive prime
divisors in polynomial arithmetic dynamics,” Integers 7(1) (2007), A26. The latter led to a paper byRichard Gottesman and Kwokfung Tang, “Quadratic recurrences with a positive density of primedivisors,” International Journal of Number Theory 6(5) (2010), 1027-1045.

Teaching Experience
At Carleton•••••••

Elliptic Curves (Math 395), Spring 2017
Calculus II (Math 121), Fall 2012, W inter 2013, W inter 2014, Fall 2014, W inter 2017
Multivariable Calculus (Math 211), W inter 2015, Fall 2017
Linear A lgebra (Math 232), W inter 2013, W inter 2014, W inter 2015, Spring 2015
Mathematical Structures (Math 236), Spring 2014, W inter 2017
Number Theory (Math 312), Spring 2013, Fall 2014
A lgebraic Number Theory (Math 395), W inter 2014

Additional courses, taught at other institutions••••

Calculus I
A bstract A lgebra I and II
Linear A lgebra and Differential Equations
Introduction to Discrete Math

Teaching-related Activities
The Delta Center for the Integration of Research, Teaching, and Learning, University of Wisconsin•

Completed a three week immersive course entitled “the college classroom”, Summer 2007. Assignments
included micro-teaching presentation, readings on mathematics pedagogy, and constructions of various
assessment mechanisms.

The Sheridan Center for Teaching and Learning in Higher Education, Brown University
Graduate Teaching Fellow, 2004-2005•

Served as one of two graduate fellows for the Center, which involved organizing and running various
teacher-training programs for graduate students on campus, including micro-teaching sessions, a New TA
Orientation, the Sheridan Teaching Seminar and Certificate Program, and a Summer Syllabus Workshop.

Teaching Consultant, 2004-2005•

Observed graduate student lectures, discussion sections, and labs, and provided feedback and constructive
criticism based on the observee’s written goals and objectives for the session.

Sheridan Center Teaching Certificate I, II, and III

6•Received these certificates after completing certain requirements, which include participation in theSheridan Teaching Seminar and related workshops, a departmental micro-teaching session, and an
individual teaching consultation.

Departmental and College Service at Carleton
Carleton Junior Faculty Affairs Committee, 2013-2015.
Carleton Faculty Compensation Committee, 2017-present
Co-organizer of the Carleton math department problem-solving group, 2012-present.
Carleton Tour of Math organizer, 2014.

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