Counting problems and semisimple groups
type: Item
instance of: scholarly article
title: Counting problems and semisimple groups
zbMATH Open document ID: 0901.11029
author: Alex Eskin
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/230497
full work available at URL: Eskin.MAN.html
review text: 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 In the third section, the author explains his recent (as yet unpublished) joinnt work with H. Masur on counting families of periodic trajectories of the polygonal billiards, which is known to be related to the problem of counting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.
reviewed by: B. Z. Moroz
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 11J25
Mathematics Subject Classification ID: 11H50
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 81Q50
zbMATH DE Number: 1184313
zbMATH Keywords: lattice points
zbMATH Keywords: ergodic theory
zbMATH Keywords: discrete subgroups of Lie groups
zbMATH Keywords: counting problems
zbMATH Keywords: symmetries
zbMATH Keywords: unipotent flows
zbMATH Keywords: asymptotic formulas
zbMATH Keywords: diophantine equations
zbMATH Keywords: diophantine inequalities
zbMATH Keywords: quantum chaos
zbMATH Keywords: billiards on rational polygons
zbMATH Keywords: quantitative Oppenheim conjecture
zbMATH Keywords: quadratic forms
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1184313