# Algebraic Geometry, Number Theory and Cryptography

## Research topics

**Algebraic geometry, cryptography, and number theory**

- Algebraic geometry
- Information and communication, circuits
- Number theory

**Algebra and topology**

- Algebraic topology
- Associative rings and algebras
- Topological groups, Lie groups

## Related websites

- Number Theory Research Group
- Institute for Security, Privacy and Information Assurance (ISPIA)
- Automorphic Representations Research Group

## Research Spotlight: Dang Khoa Nguyen

Most of my research is about algebraic dynamical systems and diophantine equations. The study of diophantine equations, which dates back to the ancient Greeks, is a branch of number theory studying integer solutions of polynomial equations. Dynamical systems are ubiquitous in modern mathematics ranging from the theory of differential equations describing the motion of celestial objects to recent advances in combinatorics using ergodic theory. Some of my recent work also involves the combinatorial and analytic aspects of the rings of polynomials over finite fields which are considered positive characteristic analogue of the ring of integers.

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.