My research interests are generally in the areas of Arithmetic Geometry, Algebraic Number Theory, and Arithmetic Dynamics. 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 mainly working on various Galois-theoretic problems in dynamics. This survey article is a good reference to learn more about the field of Arithmetic Dynamics and some of the main open problems in the area.

Publications

14. Portraits of quadratic rational maps with a small critical cycle.(arXiv)

With Tyler Dunaisky. J. Number Theory. To appear.

13. Algebraic periodic points of transcendental entire functions.(arXiv)

With Diego Marques, Carlos Gustavo Moreira, and Pavel Trojovský.
Int. J. Number Theory. To appear.

12. Dynatomic Galois groups for a family of quadratic rational maps.(arXiv)

With Allan Lacy. Int. J. Number Theory 20 (2024), no. 7, 1701-1724.

11. Quadratic points on dynamical modular curves.(arXiv)

With John Doyle. Res. number theory 10,59 (2024).

10. Morikawa's unsolved problem.(PDF)

With Jan Holly. Amer. Math. Monthly 128 (2021), no. 3, 214-237.

9. Twists of hyperelliptic curves by integers in progressions modulo p.(PDF)

With Paul Pollack. Acta Arith. 192 (2020), no. 1, 63-71.

8. A finiteness theorem for specializations of dynatomic polynomials.(arXiv)

Algebra Number Theory 13 (2019), no. 4, 963–993.

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

With Nicole Sutherland. J. Symbolic Comput. 103 (2021), 108-126.
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 187 (2018), 469-511.

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 function bdd_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.

Doctoral dissertation

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