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Group Actions Seminar

Fall 2007

Monday 3:00-4:00 pm

Tea will be served!

Oct 15 Alex Gorodnik
Princeton University/ Bristol University
Counting rational points on homogeneous varieties.
We compute the asymptotics of the number of rational points with bounded height on some homogeneous varieties. The proof uses dynamics of unipotent flows.
Oct 22 --
Oct 29 --
Nov 6 Pascal Hubert
Translation surfaces in genus 2 and 3.
I will define translation surfaces and moduli spaces of abelian differentials, SL(2,R) action. I will explain McMullen's classification of SL(2,R) orbits' closures in genus 2. I will show the existence of strange phenomena in genus 3.
Nov 12 --
Nov 19 Dubi Kelmer
Scarring on invariant manifolds for quantum maps on the torus
Quantum maps on the torus, are toy models for quantum mechanical systems with underlying chaotic classical dynamics. One of the interesting questions in the study of such systems is to classify the measures obtained as limits of quantum measures. For an (ergodic) linear map of the torus, if there is an invariant co-isotropic sub-manifold (this could only happen when the dimension is >2) then it is possible to find quantum measures that localized on this sub-manifold. In this talk I will describe this phenomenon, and show that it is also stable under certain nonlinear perturbations, thus exhibiting more "generic" examples of this scarring.
Nov 26 Emmanuel Breuillard
Ecole Polytechnique/IAS
Arithmetic heights, the Margulis Lemma and the Tits alternative
We introduce a notion of minimal height for a finite subset of matrices with coefficients in an algebraic closure of Q and show an adelic analog of the "Margulis Lemma" in this situation, which asserts that sets of small height must generate virtually solvable subgroups. This result allows to prove uniform bounds for the growth and co-growth of finitely generated subgroups of GL_d(C). We also make a connection with the Lehmer conjecture.
Dec 3 Shimon Brooks
Princeton University
Entropy of Quantum Limits for the 2-Torus
Quantized linear maps of the torus ("cat maps") are among the simpler models of quantum chaos. At present, classifying the invariant measures arising from these systems seems to be a difficult question. One interesting phenomenon (at least for the 2-torus) is a lower bound on the entropy of such measures-- equal to half of the maximal (Lebesgue) entropy. We will discuss ideas behind the proof of this fact and related results.

Archive Spring 2005