Counting problems and semisimple groups
I cite the author's abstract: ``Some natural counting problems admit extra symmetries related to actions of Lie groups. For these problems, one can sometimes use ergodic and geometric methods, and in particular the theory of unipotent flows, to obtain asymptotic formulas. We will present counting problems related to diophantine equations, diophantine inequalities and quantum chaos, and also to the study of billiards on rational polygons.'' Accordingly, the paper consists of three sections. In the first section, the recent paper [\textit{A. Eskin}, \textit{S. Mozes}, and \textit{N. Shah}, Ann. Math. (2) 143, 253-299 (1996; Zbl 0852.11054)] on counting integer points of affine homogeneous spaces of Lie groups has been surveyed. The second section, entitled ``A quantitative version of the Oppenheim conjecture'', is a brief summary of the author's joint work with \textit{G. Margulis} and \textit{S. Mozes} [Ann. Math. (2) 147, 93-141 (1998)]; following a recent preprint by P. Sarnak, the author explains, in particular, the relation of the quantitative Oppenheim conjecture on the integral values of real indefinite quadratic forms to certain eigenvalue problems inspired by quantum mechanics (``quantum chaos''). In the third section, the author explains his recent (as yet unpublished) joinnt work with H. Masur on counting families of periodic trajectories of the polygonal billiards, which is known to be related to the problem of counting cylinders of closed geodesics on the surface with a flat structure associated to the polygon in question.
scientific article; zbMATH DE number 1184313