Values of indefinite quadratic forms at integral points and flows on spaces of lattices

Abstract

This mostly expository paper centers on recently proved conjectures in two areas: A) A conjecture of A. Oppenheim on the values of real indefinite quadratic forms at integral points. B) Conjectures of Dani, Raghunathan, and Margulis on closures of orbits in spaces of lattices such as SL„(R)/SL„(Z). At first sight, A) belongs to analytic number theory and B) belongs to ergodic and Lie theory, and they seem to be quite unrelated. They are discussed together here because of a very interesting connection between the two pointed out by M. S. Raghunathan, namely, a special case of B) yields a proof of A). The first main goal of this talk is to describe the Oppenheim conjecture and various refinements and to derive them from one statement about closures of orbits in the space of unimodular lattices in R3 (see Proposition 2 in §2.3). In §§3 and 4 we put this statement in context and describe more general conjectures and results on orbit closures and invariant probability measures on quotients of Lie groups by discrete subgroups. §5 gives some brief comments on the proofs and further developments. §§6, 7, and 8 are devoted to the so-called S-arithmetic setting, where we consider products of real and p-adic groups. §6 is concerned with a generalized Oppenheim conjecture; §7 with a generalization of the orbit closure theorem proved by M. Ratner [R8]; and §8 with applications to quadratic forms. Since the subject matter of that last section has not been so far discussed elsewhere, we take this opportunity to present proofs, obtained jointly with G. Prasad. Finally, §9 gives the proof of a lemma on symmetric simple Lie algebras, a special case of which is used in §8. I am glad to thank M. Ratner and G. Prasad for a number of remarks on, and corrections to, a preliminary version of this paper, thanks to which many typos and inaccuracies have been eliminated.