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

这一建议解决了理解波在给定介质中的传输的基本挑战。这个主题非常广泛，从实验科学到理论建模。我们的重点将放在从潜在的基本物理定律推导输运方程的严格数学推导上，并因此描述在介质精细结构给出的长度尺度之上几个数量级的尺度上的影响。这项研究的令人兴奋之处在于，我们试图推导的一些输运过程是新的，将对经典的线性玻尔兹曼方程进行微妙的修正。因此，这个项目的发现不仅对数学和数学物理具有根本的兴趣，而且在线性玻尔兹曼方程作为中心模型的应用领域；例子包括辐射传递、中子散射和半导体物理。我们使用的工具建立在最近开发的theta函数的几何正则化技术上，这反过来又使用了齐次空间上群体行为的动力学。这些深入而复杂的方法的进一步发展将成为本项目的主要部分，并将在二次型分布（即缩小区间值的奥本海姆猜想的定量版本）和theta函数的值分布等长期存在的问题中具有独立的应用。

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.