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

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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.