I mainly work in the areas of arithmetic dynamics, diophantine geometry, and algebraic number theory. I'm interested in both theoretical and computational aspects of these subjects; in fact most of my work is a blend of theory and computation. At present I'm mostly focused on finding a way to prove certain special cases of a uniform boundedness conjecture in dynamics.

Publications

7. Galois groups over rational function fields and explicit Hilbert irreducibility.(arXiv)

With Nicole Sutherland. Submitted.
An implementation of the main algorithm is included in Magma via the intrinsic function HilbertIrreducibilityCurves.

6. Galois groups in a family of dynatomic polynomials.(arXiv)

J. Number Theory, to appear.

5. A local-global principle in the dynamics of quadratic polynomials.(arXiv)

Int. J. Number Theory 12 (2016), no. 8, 2265-2297.

4. Squarefree parts of polynomial values.(PDF)

J. Théor. Nombres Bordeaux 28 (2016), 699-724.

3. Computing points of bounded height in projective space over a number field.(arXiv)

Math. Comp. 85 (2016), 423-447.

2. Computing algebraic numbers of bounded height.(arXiv)

With John Doyle. Math. Comp. 84 (2015), 2867-2891.
An implementation of the main algorithm is included in Sage via the number field attribute elements_of_bounded_height.

1. Preperiodic points for quadratic polynomials over quadratic fields.(PDF)

With John Doyle and Xander Faber. New York J. Math. 20 (2014), 507-605.

Papers in preparation

  • A finiteness theorem for specializations of dynatomic polynomials.
  • Classification of preperiodic structures for quadratic polynomials over quadratic fields, with John Doyle and Joseph Wetherell.

Doctoral dissertation

Quadratic points on modular curves (PDF). Advisor: Dino Lorenzini