More specifically, some of my current research projects are:
- A sequel to the paper “D-finiteness, rationality, and height” with coauthors Jason Bell and Umberto Zannier. This paper studies the height growth of coefficients of a D-finite series; such coefficients also arise from dynamical systems related to certain birational maps on the projective spaces. Results and methods in this paper inspire the recent solution of a 55-year old conjecture of Schinzel-Zassenhaus by Vesselin Dimitrov as well as further research in dynamics by Dragos Ghioca and his coauthors.
- Proving algebraic independence of Mahler functions of polynomial type. These functions have played an important role in transcendental number theory since Mahler’s seminal work in the late 1920s. For the past 40 years, there have been intense research activities about Mahler’s functions of “linear type” due to their connection to automatic sequences. In fact, very recently, Boris Adamczewski told me that his group was successful to prove algebraic independence of such functions, hence settled a conjecture in the 1980s. On the other hand, very little is known about the algebraic independence of functions of “polynomial type”. Medvedev, Scanlon, and I are tackling this problem using ideas in our earlier papers.
- Dynamics with a positive characteristic action. A “more traditional” dynamical system involves the action of a semigroup G that is either the set of natural numbers or the ring of integers on a space X. In this project, I am interested in the behaviors of the systems when G is related to the ring of polynomials over a finite field. In fact, some questions in this setting are being investigated by my PhD student Keira Gunn.
I have a broad interest in algebra, number theory, dynamical systems, and combinatorics and I am very keen to work with students and postdocs having similar taste in these areas. For the academic year 2020-2021, I expect to have 2 new postdocs and 1 new MSc student and we aim to have a very fruitful year despite the current impact by COVID-19.