Computational Ergodic Theory, Hyperbolic Geometry and Spectral Theory

This is an application based in Mathematical Analysis, but with an emphasis on the use of ergodic theory to study applications to geometry, and areas of mathematics and the physical sciences. The strategy is to harness the power of new algorithms originating in ergodic theory to address problems in hyperbolic geometry and related fields which may depend on precise numerical values. In particular, I aim to refocus my research in a new direction to apply my expertise in computational ergodic theory to applications to (hyperbolic) geometry and spectral theory. My recent progress in related areas (for example, applications to number theory and fractal geometry using a variety of different approaches) and the techniques developed there give the prospect of deriving very high accuracy estimates, applicable to wide range of geometric problems (including those highlighted above) where it is important to have numerical values which are validated with complete rigour. Furthermore, continuing advances in computer hardware and memory storage should allow now most of the necessary computations to be be performed on a standard laptop. The novel interplay of rigorous computation and geometric problems should allow for a greater impact than more conventional methods. It is anticipated that the algorithms developed in the course of the programme would also contribute ideas to the mathematical proofs. Moreover, the techniques developed should be transferable more widely to problems in other areas, including Ergodic Theory, Dynamical Systems, Fractal Geometry and analytic number theory. The symbiosis between ergodic theory and geometry is recognised as among the most important and vibrant branches of mathematics: e.g. recent Fields medallists such as Avila, Bourgain, Lindenstrauss, Smirnov, McMullen, Yoccoz, who have been active in the interplay between these areas. The likely transformative impact of my programme has the potential to invigorate the subject in the UK, in particular by infusing it with new techniques to carry out further empirical investigations in a wide range of geometric problems where the values of numerical quantifiers are important. Often research on one topic has surprising applications in other areas. The study of resonances, one of the themes of this programme, has widespread implications far beyond geometry, dynamics or even pure mathematics. For example, they appear in the paper Rough parameter dependence in climate models and the role of Ruelle Pollicott resonances by M.Chekroun, J.Neelin, D.Kondrashov and M.Ghil, published in the prestigious Proceedings of the National Academy of Sciences. Their work describes turbulent atmospheric flows and ocean currents and their recurrent large scale patterns have low frequencies that can be formulated in terms of Ruelle-Pollicott resonances. This is illustrated in the case of the El Nino climate model. My original work was in pure mathematics (Ergodic Theory and Dynamical Systems) but has had impacts far beyond this setting, as illustrated by this example. The project will strengthen the overall UK knowledge base in the various fields concerned, notably Hyperbolic Geometry, Ergodic Theory, Algorithms and related fields, and establish establish a world-leading team in this niche of numerical validation of ergodic-theoretic computations and their applications to other fields. I have a track record of mentoring PDRAs who went on to very successful careers. The programme will fund two PDRAs who will be trained in this niche area, keep it vibrant, energized and lively, and move it forward and in new directions in the future.