To visualize your favorite fraction in your favorite base, click here (thanks to Marielle Foster and Benjamin Wedin for developing this page). For the mathematics behind these pictures, see the article "A postmodern view of fractions and reciprocals of Fermat primes" at the bottom of the list below.
To generate your very own prime number poem, click here.
The comps group I advised in winter-spring 2013 has produced a nice article discussing Shabat polynomials (that is polynomials with at most two critical values) and their links to combinatorial bicolored plane trees. It provides a supplement to an article of Shabat and Zvonkin on the same topic. The paper touches on topics as diverse as algebraic topology, Riemann surfaces, and continued fractions. Click here for the article.
Mathematical publications and preprints
Jordan Cahn, Rafe Jones, and Jacob Spear. "Powers in orbits of rational functions: cases of an arithmetic dynamical Mordell-Lang conjecture." Click here for preprint on the ArXiv.
Rafe Jones and Alon Levy. "Eventually stable rational functions." To appear in Int. J. Number Theory. Click here for preprint on the ArXiv.
Andrew Bridy, Patrick Ingram, Rafe Jones, Jamie Juul, Alon Levy, Michelle Manes, Simon Rubinstein-Salzedo, and Joseph H. Silverman. "Finite ramification for preimage fields of postcritically finite morphisms." To appear in Math. Res. Lett. Click here for preprint on the ArXiv.
Benjamin Breen, Rafe Jones, Tommy Occhipinti, and Michelle Yuen. "Wild ramification in a family of low-degree extensions arising from iteration." JP J. Algebra Number Theory Appl. 37(1) (2015), 69-104.
(Click here for preprint on the ArXiv.)
Spencer Hamblen, Rafe Jones, and Kalyani Madhu. "The density of primes in orbits of z^d + c." Int. Math. Res. Not. 2015(7), 1924-1958.
(Click here for preprint on the ArXiv.)
Rafe Jones. "Fixed-point-free elements of iterated monodromy groups." Trans. Amer. Math. Soc. 367(3) (2015), 2023-2049.
(Click here for preprint on the ArXiv.)
Robert Benedetto, Patrick Ingram, Rafe Jones, and Alon Levy. "Attracting cycles in p-adic dynamics and height bounds for post-critically finite maps." Duke Math. J. 163(13) (2014), 2325-2356.
(Click here for preprint on the ArXiv.)
Rafe Jones and Michelle Manes. "Galois theory of quadratic rational functions." Comment. Math. Helv. 89(1) (2014), 173-213.
(Click here for preprint on the ArXiv.)
Rafe Jones. "Galois representations from pre-image trees: an arboreal survey" Pub. Math. Besançon. (2013) 107-136
(Click here for preprint on the ArXiv.)
Katharine Chamberlin, Emma Colbert, Sharon Frechette, Patrick Hefferman, Rafe Jones, and Sarah Orchard. "Newly reducible iterates in families of quadratic polynomials." Involve 5(4) (2012), 481-495.
(Click here for preprint on the ArXiv.)
Rafe Jones. "An iterative construction of irreducible polynomials reducible modulo all primes." J. Algebra. 369 (2012), 114-128. (Click here for preprint on the ArXiv.)
Nigel Boston and Rafe Jones. "Settled polynomials over finite fields." Proc. Amer. Math. Soc. 140 (6) (2012), 1849-1863.
(Click here for preprint; PDF of 15 pages.)
Rafe Jones. "Achievement sets of sequences." Amer. Math. Monthly 118 (6) (2011), 508-521.
(Click here for preprint; PDF of 16 pages)
Rafe Jones and Han Peters. "Blocks of monodromy groups in complex dynamics." Geom. Dedicata 150 (1) (2011), 137-150.
(Click here for preprint on the ArXiv.)
Rafe Jones and Jeremy Rouse (with an appendix by Jeffrey D. Achter). "Iterated endomorphisms of abelian algebraic groups." Proc. Lond. Math. Soc. 100 (3) (2010), 763-794.
(Click here for preprint; PDF of 34 pages. You can also find
this preprint on the ArXiv.)
Xander Faber, Benjamin Hutz, Patrick Ingram, Rafe Jones, Michelle Manes, Thomas J. Tucker, and Michael Zieve.
"On the geometry and arithmetic of pre-images for quadratic dynamical systems." Math. Res. Lett.16 (1) (2009), 87-101.
(Click here for the preprint
version on the ArXiv; PDF of 15 pages.)
Nigel Boston and Rafe Jones. "The image of an arboreal Galois representation."
Pure and Applied Mathematics Quarterly5 (1) (Special Issue: in honor of Jean-Pierre Serre, Part 2 of 2) (2009), 213-225.
(Click here for preprint; PDF of 13 pages)
Rafe Jones. "The density of prime divisors in the arithmetic dynamics of quadratic polynomials."
J. Lond. Math. Soc. (2) 78 (2) (2008), 523--544.
(Click here for preprint version (some differences with published version); PDF of 23 pages.
You can also find this preprint version on the ArXiv.)
Rafe Jones. "Iterated Galois towers, associated martingales, and the p-adic Mandelbrot set."
Compositio Math. 43 (5) (2007), 1108-1126.
(Click here for preprint version; PDF of 23 pages)
Nigel Boston and Rafe Jones. "Arboreal Galois representations."
Geom. Dedicata. 124(1) (2007), 27-35 (electronic)
(Click here for preprint version; PDF of 12 pages)
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
Rafe Jones and Jan Pearce. "A postmodern view of fractions and the reciprocals of Fermat primes."
Math. Mag. 73 (2000), 83-97.
Click here for preliminary version with figures omitted; PDF of 18 pages. Also
click here to see our pictures of fractions on
the cover of the issue where the article appeared.
Other publications and writing
"Numbers: A Very Short Introduction--A Book Review" appeared in the January 2012 issue of the Notices of the AMS. Click here to go to the article (PDF, 2 pages)
My dissertation is entitled "Galois Martingales and the Hyperbolic Subset of the p-adic Mandelbrot Set." You can see the introduction (PDF, 6 pages) or the whole thing (PDF, 90 pages of actual content). Note that to get the bibliography for the intro you'll have to look at the last three pages of the whole thing
"Ninety-One Years," [Poem] appeared in New Millenium Writings, June 2004. Click for the poem (PDF, 1 page)
"In Code: A Mathematical Journey--A Book Review" appeared in the April 2003 issue of the Notices of the AMS. Click here to go to the article (PDF, 2 pages)
"Stern Perfection: Mathematics as a Fine Art," appeared in Brown University's
Catalyst Special Edition , 2002. Click for the article (PDF, 3 pages)
"A Summer at Discovery.com; appeared in the June/July 2002 issue of the
Notices of the AMS .
Click here
to go to the article (PDF, 3 pages)