Second meeting, Tel Aviv University, 10.1.16, Schreiber building
Schreiber room 8
Uniform exponential escape of random matrix products from algebraic subgroups
and a generalization of Leighton's combinatorial common covering theorem.
We shall discuss a general non-concentration result for random matrix
products, which is a key ingredient in the vast literature on super-strong approximation following the famous work of Bourgain-Gamburd.
While this has always been a very technical step, our entirely different approach
is much softer, and covers also, for the first time, the case of positive
characteristics. A special role in the proof is played by a spectral extension
of a famous theorem of Leighton: if two finite graphs admit the same universal covering (tree), then they admit a finite common covering. A Riemannian version of this result is discussed as well.
Joint work with Asaf Hadari.
Schreiber room 8
A strengthened version of Banach property (T) and angles between projections
This lecture has two aims: first, to introduce a new definition of a strengthened version of Banach property (T) inspired by V. Lafforgue’s work on strong Banach property (T). Second, to present a criterion for this version of Banach property (T) based on the idea of angles between projections.
Affine symmetries of translation surfaces
In this talk we will discuss what we know about affine symmetries of
translation surfaces. We will put special interest in the linear parts of
these symmetries when the translation surface has infinite topological type.
Schreiber room 7
Diffraction theory for aperiodic point sets in Lie groups
The study of aperiodic point sets in Euclidean space is a classical topic in harmonic analysis, combinatorics and geometry. Aperiodic point sets in R^3 are models for quasi-crystals, and in this context it is of interest to study their diffraction measure, i.e. the way they scatter an incoming laser or x-ray beam.
By a classical theorem of Meyer, every sufficiently regular aperiodic point set in a Euclidean space is a shadow of a periodic one in a larger locally compact abelian group. The diffraction of these "model sets" can be computed in terms of a certain groups of irrational rotations of an associated torus.
In recent work with Bjoerklund and Pogorzelski we free model sets from the unnatural corset of abelian groups and consider them in their natural generality, i.e. in the context of arbitrary locally compact groups. This leads to interesting new examples such as (S-)arithmetic quasi-crystals in real and p-adic Lie groups and Burger-Mozes type quasi-crystals in automorphism groups of trees. We develop a general diffraction theory for such point sets, and relate their (spherical) diffraction to the (spherical part of the) automorphic spectrum of the underlying lattice.
In our general context, the irrational rotation of the torus gets replaced by an "almost homogeneous dynamical system", i.e. an action of a group G on a space of the form (GxH)/\Gamma for a lattice \Gamma in GxH. The study of diffraction of model sets thus motivates a new kind of "almost homogeneous dynamics", and we will sketch some very modest beginnings of such a theory.
You are cordially invited.