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Homogeneous Flows and their Application in Kinetic Theory (HFAKT)
Start date: Apr 1, 2012, End date: Mar 31, 2017 PROJECT  FINISHED 

"Since the pioneering work of Maxwell and Boltzmann in the 1860s and 1870s, a major challenge in mathematical physics has been the derivation of macroscopic evolution equations from the fundamental microscopic laws of classical or quantum mechanics. The key idea of the present proposal is to introduce a renormalization technique that will provide a new route to attack some of the most important questions in the kinetic theory of gases. This technique uses the ergodic theory of flows on homogeneous spaces (homogeneous flows for short), and builds on my recent breakthrough in the case of the Lorentz gas (in joint with Andreas Strömbergsson, Uppsala). The key feature of the proposed approach is measure rigidity, a deep and powerful technical tool that has so far mainly seen success in solving long-standing problems in number theory and quantum chaos. The arguments developed in this proposal are not only interesting from a rigorous mathematical viewpoint, but also yield a heuristic mechanism for finding previously unknown kinetic transport equations that incorporate the effects of long-range order in microscopic particle distributions. This project joins together two distinct research fields, kinetic theory and the ergodic theory of flows on homogeneous spaces. If successful, the proposed research will constitute a significant breakthrough in the subject."
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