Asymptotiska problem på homogena rum, med tillämpningar i talteori och matematisk fysik
Tidsperiod: 2012-01-01 till 2015-12-31
Projektledare: Andreas Strömbergsson
Budget: 4 050 000 SEK
The proposed research is a multifaceted investigation centering on problems in homogeneous dynamics and fundamental questions about lattices. We will focus in particular on problems which hold the promise of important applications in the kinetic theory of gases or in number theory. The proposed work features a rich interplay in technique.One of our long-term goals is to shed new light on the problem of giving a microscopic justification of the Boltzmann equation for the hard-sphere gas model; this is an outstanding problem in mathematical physics. Another long-term goal concerns a central and long-standing problem in the geometry of numbers: to prove sharper bounds on the maximal density of a lattice sphere packing in Euclidean space in large dimension. Some of the problems which we will consider involve proving precise rate of convergence results for special cases of Ratner´s theorem on unipotent flows, often with several number theoretical applications. Yet other problems concern establishing new kinetic transport equations that describe the Boltzmann-Grad limit of the Lorentz gas for scatterer configurations which are not periodic but still exhibit long-range order.