## Final Report Summary - HFAKT (Homogeneous Flows and their Application in Kinetic Theory)

The Lorentz gas is one of the simplest, most widely used models to study the transport properties of rarified gases in matter. It describes the dynamics of a cloud of non-interacting point particles in an infinite array of fixed spherical scatterers. A major challenge of this research project was to understand the nature of the kinetic transport equation that governs the macroscopic particle dynamics in the limit of low scatterer density (the Boltzmann-Grad limit). Lorentz suggested in 1904 that this equation should be the linear Boltzmann equation. This was confirmed in three celebrated papers by Gallavotti, Spohn, and Boldrighini, Bunimovich and Sinai, under the assumption that the distribution of scatterers is sufficiently disordered. A key finding of this research project, obtained in collaboration with A. Strombergsson (Uppsala University), is that in the case of strongly correlated scatterer configurations (for example the vertex set of a Penrose tiling), the macroscopic dynamics is described by a Markovian random flight process, whose transport equation is in general not the linear Boltzmann equation. The description of the limit process requires the introduction of hidden variables, which depend on the class of scatterer configuration. In the case of a Penrose tiling the limit process is, for example, determined by a random cut-and-project set, which is stationary under the action of a Hilbert modular group. A particularly striking feature in the case of the periodic Lorentz gas is a heavy tail for the distribution of free path lengths, with a diverging second moment. In this case we established, in joint work with B. Toth (Bristol & Budapest), that this divergence leads to a central limit theorem for particle transport in the limit of large times t, with a t log t divergence of the mean-square displacement. This is the first rigorous proof of superdiffusion in the Lorentz gas in dimension three or higher. The key progress in this research project has been achieved through the development and application of sophisticated methods in the theory of homogeneous flows, which are used to "renormalize" the gas dynamics. In particular, we have derived equidistribution theorems for translates of spheres, horospheres and more general embedded submanifolds in various moduli spaces through the application of Ratner's theorem. In addition to applications in kinetic theory, we have exploited these techniques to answer a number of open questions in other research areas, including the theory of Diophantine approximation, pseudo-randomness in the distribution of number-theoretic sequences modulo one, and the distribution of directions in hyperbolic lattices and quasicrystals.