The Weizmann Institute of Science Faculty of Mathematics and Computer Science Mathematical Analysis and Applications Seminar Lecture Hall, Room 1, Ziskind Building on Tuesday, June 14, 2011 11:00 - noon Uri Shapira ETH Zurich will speak on Homogeneous orbit closures and applications Abstract: I will discuss a recent joint work with Elon Lindenstrauss: One of the main open problems in homogeneous dynamics is to classify the ``nice" orbit closures for the action of the diagonal group on the space of three dimensional lattices. We present a variety of concrete lattices for which we know the orbit closure is nice, as well as lattices with funny orbit closures. Some non-trivial consequences to Diophantine approximations of algebraic numbers follow from our analysis. Ratner's celebrated theorem sais basically that the orbit closure of any point in a homogeneous space G/Gamma under the action of a group generated by unipotents is a homogeneous object. This theorem has numerous applications in number theory. To the opposite extreme we study the action of the diagonal group (having no unipotents at all) and prove that for a variety of concrete points, the conclusion of Ratner's theorem still holds. Non-trivial applications to number theory will be described as well (in the spirit of Littlewood's conjecture). Mathematical Analysis and Applications Seminar Web Page: <http://www.wisdom.weizmann.ac.il/~vered/analsem.html> --------------------------------------------------------- Technion Math Net-2 (TECHMATH2) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Diana Mandelik <diana.mandelik@weizmann.ac.il>