Wave transport in low-density matter, Siegel theta functions, and homogeneous flows

This proposal addresses the fundamental challenge of understanding wave transport in a given medium. The subject is extremely broad, ranging from experimental science to theoretical modelling. Our focus will be on the rigorous mathematical derivation of transport equations from the underlying fundamental laws of physics, and to thus describe effects on scales which are several orders of magnitude above the length scale given by the fine structure of the medium. The exciting aspect of the proposed research is that some of the transport processes we seek to derive are new and will expose subtle corrections to the classical linear Boltzmann equation. The findings of this project will thus not only be of fundamental interest in mathematics and mathematical physics, but also in applied areas where the linear Boltzmann equation serves as a central model; examples include radiative transfer, neutron scattering and semiconductor physics. The tools we employ build on recently developed techniques on the geometric regularisation of theta functions, which in turn uses the dynamics of group actions on homogeneous spaces. The further development of these deep and sophisticated methods will form a major part of this project, and will have independent applications in long-standing questions on the distribution of quadratic forms (i.e. quantitative versions of the Oppenheim conjecture for values in shrinking intervals) and the value distribution of theta functions.

The nonlinear and linear Boltzmann equations serve as fundamental phenomenological models for a large number of physical applications, including semiconductor physics and radiation damage calculations. Their derivation from microscopic Newtonian or quantum laws is a long-standing problem in mathematical analysis. The exciting aspect of this proposal is that the microscopic laws in crystals or quasicrystals produce new, unexpected macroscopic transport equations, and there is every expectation that the outcomes of this project will have global impact in both the mathematics and physics communities. The PI has a proven track record of publishing in top general mathematics periodicals and specialist journals in number theory, ergodic theory, as well as mathematical and theoretical physics.

In addition to the project's academic impact described above (cf. Academic Beneficiaries), this project will make a significant contribution to the UK's competitiveness and leadership in highly topical research fields. Through its position at the interface of analysis, ergodic theory and number theory, the proposed project furthermore addresses the recommendations of the 2010 International Review of Mathematical Sciences in the three areas (see Case for Support/National Importance for details). Indeed, the interface of analysis, ergodic theory and number theory has seen some spectacular advances in recent years, some of which were recognised by three Fields medals (Lindenstrauss 2010, Avila 2014, Venkatesh 2018).

The project lends itself ideally to public engagement activities. Chaos, quantum physics, and random patterns in simple number systems are topics that can excite a lay audience. The proposed research unifies these subjects in a natural way. The PI has a proven record in public engagement, including several Cafe Scientifique talks with lay audiences and lectures for school children. The project team will develop jointly public engagement activities, which will include lectures, a website and a professionally produced 10min youtube video. The video will explain to a lay audience the fundamental science behind the project and convey the excitement of taking part in ground-breaking research. The team will receive training and support from the host institution's Public Engagement team (www.bristol.ac.uk/public-engagement).