University/ Bristol University
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.
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.
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.
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
analog of the "Margulis Lemma" in this situation, which asserts that
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
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