EPSRC DTP PhD studentship: Number Theory in Function Fields and Random Matrix Theory

University of Exeter - College of Engineering, Mathematics and Physical Sciences

Main supervisor: Dr. Julio Andrade (University of Exeter)

The proposed project is in number theory, an area of pure mathematics which is concerned with prime numbers and solutions to equations. It has long been understood in the field that there are strong analogies between number theory and the geometry of curves. As a deeper understanding of this analogy could have tremendous consequences, the search for a unified approach and the reinterpretation of geometric methods has been a source of inspiration and success for number theorists.

There is a different setting, in addition to the realm of the integer numbers, where we can study number theory, namely function fields. The main idea is that instead of studying integers and their properties we study polynomials over finite fields - a finite field is a set of numbers that behaves like the set of real numbers but contains a finite number of elements. One of the reasons to study polynomials over finite fields is that the analogies between the integers and such polynomials are striking and that by finding solutions to the function field analogue problem can lead to solutions to the classical case.

This project aims to investigate some of the analogies between numbers and polynomials by using techniques which have already been successful either for numbers or for polynomials, and by reinterpreting the geometry of curves in arithmetic terms. This will hopefully help to build a unifying theory that encapsulates these two parallel worlds. The focus of this research is to investigate a new way to tackle many simple but still unanswered questions about integers: for example, the distribution of the prime numbers, the number of divisors of different integers, and the relationship of number theory with physics. Previous researchers have studied a class of number theory questions that can be re-expressed in a form that makes them amenable to attack using techniques first developed by mathematical physicists to analyse complex quantum mechanical systems, such as large atomic nuclei and electrons subject to random forces.

The student will benefit from this project in several ways. First, he will have contact with advanced mathematics that is not seen in undergraduate courses. Second, he will be able to develop his skills in two or more areas of mathematics including analytic number theory, function fields, algebraic geometry and the theory of the Riemann zeta function. During his PhD in this project, the student also will be able to give talks at seminars in the UK and abroad and to attend conferences since this is a hot topic in number theory nowadays. The student will also be able to interact with other mathematicians here in Exeter but also in Bristol, Oxford and in the USA and so build some network that will be essential for job applications.