Dr Corinna Ulcigrai is researching in Teichmueller dynamics and ergodic theory, a branch of pure mathematics which studies the chaotic properties of dynamical systems, that is, systems such as a moving particle, which evolve according to a deterministic law but whose long-term behaviour is hard to predict. Dynamical systems which are researched in Teichmueller dynamics include polygonal mathematical billiards. Here, analogously to an actual game of billiards, mathematicians study a ball travelling without friction at a constant speed that never hits a pocket and rebounds infinitely off the table walls. Dr Ulcigrai is tackling unanswered questions on these and other slowly chaotic systems, known as parabolic dynamical systems, using a sophisticated technique called renormalisation, which allows her to magnify chaotic behaviour occurring on very small scales. This groundbreaking research has connections with different areas of physics and mathematics, such as number theory and algebraic geometry.

Based in the School of Mathematics, renowned for world-leading groups in ergodic and number theory, Dr Ulcigrai is one of a few experts in the UK researching in Teichmueller dynamics. It is an extremely topical and challenging area, one which continues to engage three Fields medallists, the equivalent of Nobel prize winners in mathematics. She has established collaborations with a worldwide network of leading scientists and following an Engineering and Physical Sciences Research Council (EPSRC) grant, she is drawing a stream of these high calibre visitors to Bristol.

She has made headway with one collaborator, Professor John Smillie (Cornell University). By using the renormalisation technique, they described fully a class of sequences that arise from recording the order of sides hit by a billiard trajectory in a regular polygon shaped table. This new class displays many similarities with the well-known Sturmian sequences which describe how a line intersects a square grid. It is believed that the ancient Greeks may have already observed the patterns described by Sturmian sequences when attempting to predict the lunar cycles and create ancient calendar systems.

Dr Ulcigrai also made history when she established certain chaotic properties for flows on surfaces originally introduced by Sergei Novikov in the 1980s to describe the trajectories of an electron under a magnetic field. She settled the generalized version of a 20-year-old conjecture, a mathematical open problem, suggested by Vladimir Arnold (1990), a renowned Russian mathematician, regarding whether these flows had the highly chaotic mathematical property known as mixing. Over the last four years she has published this work in a series of papers, the most recent of which appeared in the Annals of Mathematics, one of the most prestigious publications in pure mathematics.

She also recently made a breakthrough in a spin-off project with Polish collaborator, Professor Krzysztof Frączek (Nicolaus Copernicus University) in another pioneering area, infinite periodic billiards. Examples of billiards that they studied are infinite tubes where a billiard trajectory bounds off periodically spaced barriers and a planar billiard with periodically spaced rectangular obstacles. The latter, known as the Ehrenfest billiard, was introduced in 1912 to model a gas particle. Frączek and Ulcigrai proved that these two systems are not as chaotic as previously thought which came as a surprise to the mathematicians working in this field.

Dr Ulcigrai says: “The ergodic theory group in Bristol is young and dynamic and is closely linked to the number theory and quantum chaos groups. The study of mathematical billiards and, more generally, Teichmueller dynamics is a fascinating and beautiful subject. Mathematicians study abstract spaces - so called ‘moduli spaces’ - which are intrinsically beautiful in themselves and have important applications in other research areas. Usually they are exploited to deduce very precise chaotic properties of certain typical physical systems. Recently we developed a renormalization technique which also works for (some) non-typical systems.”