Research Fellows Directory
Professor Jonathan Keating FRS
University of Bristol
I am interested in connections conjectured to hold between the distribution of the Prime Numbers and the mathematical description of complex quantum systems. Understanding the distribution of the primes is related to some of the most profound challenges in Mathematics, including, for example, the Riemann Hypothesis. Over the past 40 years, considerable evidence has accumulated pointing to a deep underlying connection with the area of Mathematical Physics that describes how the quantum energy levels in complex systems are distributed. This area is called Random Matrix Theory. Examples of complex quantum systems include large atoms and molecules, nuclei, electron transport through a random environment, lasers etc. Some of the evidence we have comes from numerical experiments and some from special cases when we can prove the link. However, we are very far from understanding the connection in general: the extent to which it is true, and if so, the fundamental reasons why. It is believed that substantial progress in this direction would have significant implications for Prime Number Theory.
My current focus is on a class of examples for which this connection is proved to hold. I am exploring some long-standing problems about the primes and other arithmetically interesting numbers in this context. The aim is to express these problems in a form that will make them amenable to the techniques coming from Random Matrix Theory and hence to resolve them.