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>