Beyond the zeta zeroes

This project is on the dynamical zeta function, whose role is to count periodic orbits of a dynamical system or a flow. In 1859 Riemann introduced his now famous zeta function to study properties of the prime numbers, which plays an important role in analytic number theory. One of its applications is the proof of the Prime Number Theorem. The construction of the zeta function has since been generalised to a variety of settings, and associated with counting problems in arithmetic, algebra, dynamical systems, geometry and spectral graph theory. Just like their famous prototype, zeta functions carry information about properties of the underlying object; for instance, in uniformly hyperbolic settings the largest leading zero of the dynamical zeta function is related to the Hausdorff dimension of the limit set via the Bowen--Ruelle formula. 

Inspired by the influential thermodynamic formalism theory, the present project aims: (1) To develop numerical methods for locating the zeroes of various zeta functions; (2) To study connections between properties of the zero set and key characteristics of underlying objects aiming to explain phenomena observed empirically; and (3) To explore similarities between zeta functions arising in different settings.
The zeta functions arising in seemingly different scenarios tend to have some features in common: meromorphic extension to the complex plane, location of poles and zeros, the Euler product formula, and functional equation. This project offers an exciting opportunity to learn and experience the interplay of several branches of pure mathematics and to contribute to this blooming research area.

Open to any UK or international candidates. Up to 30% of our UKRI funded studentships can be awarded to candidates paying international rate fees.

You will need to meet the minimum entry requirements for our PhD programme.

Applicants should be on track to receive (or have been awarded) a First Class or good Upper Second Class UK Honours degree (or equivalent) in Mathematics. A Masters level qualification would be an advantage.