Faculty of Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics

# Project B1

## Asymptotic distributions, lattices and groups

Principal Investigator(s) Other Investigators

## Summary:

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.

## Recent Preprints:

 13065 Coarse topological transitivity on open cones and coarsely J-class and D-class operators PDF | PS.GZ 13062 Arithmetic Groups PDF | PS.GZ 13032 Non-planarity and metric Diophantine approximation for systems of linear forms PDF | PS.GZ 13023 Large cycles and a functional central limit theorem for generalized weighted random permutations PDF | PS.GZ 13021 The Auslander conjecture for dimension less than 7 PDF | PS.GZ 13016 Strongly solvable spherical subgroups and their combinatorial invariants PDF | PS.GZ 13015 On quotients of affine spherical varieties by unipotent subgroups PDF | PS.GZ 13014 Dynamics of perturbations of the identity operator by multiples of the backward shift on $l^\infty(\mathbb N)$ PDF | PS.GZ 13005 Recurrent Linear Operators PDF | PS.GZ 13003 Distribution of values of quadratic forms at integral points PDF | PS.GZ