My research interests are in arithmetic dynamical systems, invariant theory and moduli spaces, and *p*-adic analysis and geometry. My email is alonlevy@kth.se.

Below are links to preprints of my papers. If they conflict with the arXiv citations, the versions here are more current.

*The Space of Morphisms on Projective Space*, Acta Arithmetica 146 (2011), 13-31*The Semistable Reduction Problem for the Space of Morphisms on*, Algebra and Number Theory 6 No. 7 (2012), 1483-1501**P**^{n}*Attracting Cycles in p-adic Dynamics and Height Bounds for Post-Critically Finite Maps*, joint with Rob Benedetto, Patrick Ingram, and Rafe Jones, Duke Math Journal 163 No. 13 (2014), 2325-2356*An Algebraic Proof of Thurston's Rigidity for Maps With a Superattracting Cycle*, submitted*Uniform Bounds for Pre-Periodic Points in Families of Twists*, joint with Michelle Manes and Bianca Thompson, Proceedings of the American Mathematical Society 142 (2014), 3075-3088*Isotriviality and the Space of Morphisms on Projective Varieties*, joint with Anupam Bhatnagar, submitted*The McMullen Map in Positive Characteristic*, submitted*Finite Ramification for Preimage Fields of Postcritically Finite Morphisms*, joint with Andrew Bridy, Patrick Ingram, Rafe Jones, Jamie Juul, Michelle Manes, Simon Rubinstein-Salzedo, and Joseph Silverman, submitted*Isolated Periodic Points in Several Nonarchimedean Variables*, submitted*Eventually Stable Rational Functions*, joint with Rafe Jones, to appear in International Journal of Number Theory (2016).

One more paper is in pre-preprint form; text is available to colleagues upon request:

*PCF All the Way Down*, joint with Tom Tucker. We prove that if*f*:*X*-->*X*is postcritically finite, then the restriction of the appropriate iterate of*f*to any periodic hypersurface is also postcritically finite. We use this to strengthen the main theorem in Bridy et al.

Medium-term projects, with submission expected by early 2017:

*Eventually Stable Quadratic Functions*, joint with John Doyle and Tom Tucker. We use the eventual stability result above of myself and Jones to prove that quadratics defined over a function field of characteristic zero are eventually stable.*Berkovich Space in Several Variables*, joint with Tyler Foster. We come up with an explicit description of a large slice of Berkovich**P**^{n}, suitable for doing dynamics, by analyzing all the possible images of polydisks under morphisms from**P**^{n}to itself. Our tools include tropical geometry and higher-dimensional rigid geometry.

Longer-term projects:

*The*abc*Conjecture and Arboreal Representations*, joint with Rafe Jones. We define the arboreal representation to be the action of the absolute Galois group*G*on the infinite backward image tree of_{K}*a*in*K*under*f*:**P**^{1}_{K}-->**P**^{1}_{K}. Conjecturally, it acts on the tree with finite-index image when the critical orbits of*f*are infinite and independent. We study this question under the assumptions that the*abc*conjecture holds and that*f*is eventually stable.*Attracting Cycles in Higher-Dimensional Non-Archimedean Dynamics*. I use higher-dimensional rigid geometry to generalize the attracting cycles result with Benedetto, Ingram, and Jones to self-morphisms of**P**^{n}. Key tools include my in-preparation papers above with Tucker and with Foster.

2016-11-12 18:05 CET