I like algebraic number theory, heights, and diophantine geometry. What does this mean?

I am interested in the solutions of polynomial equations in one variable (the theory of algebraic numbers), and in the algebraic solutions to polynomial equations in several variables (diophantine geometry). I am especially interested in heights, which are real-valued functions that measure the complexity of an algebraic number or an algebraic point on a variety. Most of my research has focused on studying properties of heights (especially lower bounds) over infinite extensions of the rational numbers. This is motivated by applications to diophantine equations, where a powerful tool is to compare height estimates to prove that equations don’t have too many solutions. I’m also interested in lower bounds for heights for their own sake, and especially in Lehmer’s Problem.

I also like: elliptic curves, Galois theory, group theory, arithmetic statistics, computing things (SAGE, GAP, MAGMA).

Here is the long version of my research statement.

My papers and preprints:

Remarks on Rémond’s generalized Lehmer problems
submitted
preprint
Multiplicative approximation by the Weil height
(with J. D. Vaaler)
submitted
preprint
Slicing the stars: Counting algebraic numbers, integers, and units by degree and height
(with Joseph Gunther)
Algebra and number Theory, 11(6):1385-1436, 2017.
preprint (revised) arXiv
Relative Bogomolov extensions
Acta Arithmetica 170 (2015), no. 1, 1-13.
preprint (revised) arXiv
Small points and free abelian groups
(with Philipp Habegger and Lukas Pottmeyer)
International Mathematics Research Notices 2015, no. 20, 10657-10679.
arXiv
On the compositum of all degree d extensions of a number field
(with Itamar Gal)
J. Théor. Nombres Bordeaux 26 (2014) no. 3, 655-672.
preprint arXiv
Congruences for Ramanujan’s f and ω functions via generalized Borcherds products
(with Jennifer Berg, Abel Castillo, Vítězslav Kala, Richard Moy, and Chongli Wang — project from the 2013 Arizona Winter School)
Ramanujan J. 35 (2014), no. 2, 327-338.
preprint arXiv

Here are some tables I made which give the orders and minimal permutation degrees for all of the finite simple groups of Lie type. I could not find this information all in one place.