GPS Applied Mathematics Seminar
All seminars are on XXX at X:XX pm in Van Vleck XXX, unless otherwise indicated.
|Sep 9||Manfred Einsiedler (ETH-Zurich)||Periodic orbits on homogeneous spaces||Fish|
Fri, Sept 9: Manfred Einsiedler (ETH-Zurich)
Periodic orbits on homogeneous spaces
We call an orbit xH of a subgroup H<G on a quotient space Gamma \ G periodic if it has finite H-invariant volume. These orbits have intimate connections to a variety of number theoretic problems, e.g. both integer quadratic forms and number fields give rise periodic orbits and these periodic orbits then relate to local-global problems for the quadratic forms or to special values of L-functions. We will discuss whether a sequence of periodic orbits equidistribute in Gamma \ G assuming the orbits become more complicated (which can be measured by a discriminant). If H is a diagonal subgroup (also called torus or Cartan subgroup), this is not always the case but can be true with a bit more averaging. As a theorem of Mozes and Shah show the case where H is generated by unipotents is well understand and is closely related to the work of M. Ratner. We then ask about the rate of approximation, where the situation is much more complex. The talk is based on several papers which are joint work with E.Lindenstrauss, Ph. Michel, and A. Venkatesh resp. with G. Margulis and A. Venkatesh.