John R. Doyle    

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Research Interests

My main research interests lie in the areas of arithmetic dynamics and algebraic number theory. Generally speaking, arithmetic dynamics is the study of the dynamics of rational maps defined over fields of arithmetic interest, like number fields and p-adic fields. Much of my work has involved studying properties of certain algebraic curves known as dynamical modular curves, which parametrize maps together with points exhibiting certain behaviors under iteration by those maps. I'm also interested in rational dynamics on the Berkovich projective line.

Preprints and published articles are available on the arXiv
(Note that the arXiv version may differ slightly from the published version)

Papers in preparation
[13] Quadratic polynomials over function fields are eventually stable (with Alon Levy and Thomas Tucker)
[12] Classification of preperiodic structures for quadratic polynomials over quadratic fields (with David Krumm and Joseph Wetherell)

[11] Preperiodic points for quadratic polynomials over cyclotomic quadratic fields
Preprint. (arXiv)
[10] Gonality of dynatomic curves and uniform boundedness of preperiodic points (with Bjorn Poonen)
Submitted. (arXiv)

[9] Dynamical modular curves for quadratic polynomial maps
Trans. Amer. Math. Soc., to appear. (journal | arXiv)
[8] Reduction of dynatomic curves (with Holly Krieger, Andrew Obus, Rachel Pries, Simon Rubinstein-Salzedo, and Lloyd West)
Ergodic Theory Dynam. Systems (2018; published electronically). (journal | arXiv)
[7] Preperiodic points for quadratic polynomials with small cycles over quadratic fields
Math. Z. (2017; published electronically). (journal | arXiv)
[6] Preperiodic portraits for unicritical polynomials over a rational function field
Trans. Amer. Math. Soc. 370 (2018), no. 5, 3265–3288. (journal | arXiv)
[5] Configuration of the crucial set for a quadratic rational map (with Kenneth Jacobs and Robert Rumely).
Res. Number Theory 2 (2016), 2:11. (journal | arXiv)
[4] Preperiodic portraits for unicritical polynomials
Proc. Amer. Math. Soc. 144 (2016), no. 7, 2885–2899. (journal | arXiv)
[3] Computing algebraic numbers of bounded height (with David Krumm)
Math. Comp. 84 (2015), no. 296, 2867–2891. (journal | arXiv)
[2] Preperiodic points for quadratic polynomials over quadratic fields (with Xander Faber and David Krumm).
New York J. Math. 20 (2014), 507–605. (journal | arXiv)
[1] Apollonian circle packings of the half-plane (with Michael Ching)
J. Comb. 3 (2012), no. 1, 1–48. (journal | arXiv)

[0] Dynamics of quadratic polynomials over quadratic fields (pdf; 210 pages)
The results of my thesis [will] appear in the three articles [7], [9], and [11], and the interested reader is encouraged to read those instead. The main purpose of posting my thesis here is that the details of certain computations are omitted in [7], but they appear in full in my thesis.

If you decide to peruse my thesis, please be aware that there are some minor errors (though nothing that invalidates the main results) that will have been fixed for publication. Also note that some of the notation — specifically, the names of the dynamical modular curves — has since been changed and will appear differently in the published articles. Finally, some of the results have since been strengthened. In particular, the proof of Conjecture 2.34 will appear in [9].