Asymptotic distributions, lattices and groups
In this project F. Götze will study the distribution of definite as well as indefinite forms of second and higher order on lattices in connection with lattice point problems, diophantine inequalities and the so-called quantum chaos problem. The methods developed here apply as well to approximation results for nonlinear statistics of random variables in probability theory. The investigation of indefinite forms leads to dynamical and geometric problems for linear algebraic groups as well as for their arithmetic and geometrically relevant discrete subgroups and the corresponding homogeneous spaces which will be studied by H.~Abels. Here the focus is on problems by Auslander, Milnor and Siegel and the geometry of reductive groups. Joint research efforts will be devoted to the study of generic and stochastic distribution properties of eigenvectors as well as to open problems related to effective bounds for the quantitative Oppenheim-conjecture.
Several of the research topics mentioned will be studied in collaboration with G.A. Margulis.