My primary field of research is Arithmetic Dynamics, which is the study of number theoretic properties of dynamical systems. In particular, I study number fields known as *iterated extensions*, which are named for their construction. Namely, if *f* is a monic polynomial with integer coefficients, and *f ^{n}* denotes the

Recently I have been looking at special points on hyperelliptic curves. A point *P* on an algebraic curve *C* is a Weierstrass point if the space of functions defined over *C* with poles restricted to *P* is larger than what would be expected by the Riemann-Roch theorem. The differences in the dimensions of these spaces (counted by the maximum order of the pole at *P*) are realizable as a numerical semigroup. Thus by understanding the semigroup, we may classify points on the curve.

I also have an interest in unit groups, and in particular, elliptic units. Following work of Greene and Hajir, I have written a program in PARI/GP which computes an "ideal" generating set of elliptic units in unramified extensions of imaginary quadratic fields. The code is available here: [code]. (There are a few known bugs, so contact me before using.)

- [link] with Michael Urbanski, Index divisibility in the orbit of 0 for integral polynomials (to appear in
*Integers*). - [link] with Hanson Smith and Katherine Stange, A family of monogenic
*S*quartic fields arising from elliptic curves,_{4}*J. Number Theory*, 197:361--382, 2019. - [link] with Caleb Shor, Characterization of numerical semigroup complements via Apéry sets.
*Semigroup Forum*, 1--17, 2018. - [link]
with Annie Chen and Katherine Stange, Index divisibility in dynamical sequences and cyclic orbits modulo
*p*.*New York J. Math.*, 23:1045--1063, 2017. - [link]
with Caleb Shor, On Sylvester sums of compound sequence semigroup complements.
*J. Number Theory*, 180:45--72, 2017. - [link]
A note on the monogeneity of power maps.
*Albanian J. Math*, 11(1):3--12, 2017. - [link]
Discriminants of simplest 3
^{n}-tic extensions.*Funct. Approx. Comment. Math.*, 53(2):193--214, 2015. - [link]
Discriminants of Chebyshev radical extensions.
*J. Théor Nombres Bordeaux*, 26(3):607--633, 2014. - [link]
Chebyshev action on finite fields.
*Disc. Math.*, 315:83--94, 2014. - [link]
with Aaron Yeager, Characterization of the vertex-reinforced random walk and trapping subgraphs.
*The Pentagon: A Mathematics Magazine for Students*, 68(1):21--28, 2008.