Conférence Centre International de Rencontres Mathématiques

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1712

ECOLE DE RECHERCHE

Espaces homogènes. Approximation diophantienne. Mesures stationnaires  (Ecole thématique du CNRS)
6 – 10 février, 2017

La dynamique homogène est l’étude des propriétés asymptotiques de l’action de sous-groupes de groupes de Lie sur leurs espaces homogènes. L’étude de ces flots homogènes a attiré une attention considérable au cours des cinquante dernières années. Ce sujet inclut de nombreux exemples concrets  de systèmes dynamiques de nature  géométrique comme les difféomorphismes d’Anosov du tore ou encore et les flots géodésique sur les variétés à courbure négative. La théorie trouve de nombreuses applications dans l’étude des flots géodésiques et horocycliques sur les espaces de modules.
Comité scientifique & Comité d’organisation

Boris Adamczewski (Aix-Marseille Université)
Jayadev Athreya (Washington University Seattle)
Paul Mercat (Aix-Marseille Université)
Frédéric Palesi (Aix-Marseille Université)
« We the organizers of this conference affirm that scientific events must be open to everyone, regardless of race, sex, religion, national origin, sexual orientation, gender identity, disability, age, pregnancy, immigration status, or any other aspect of identity. We believe that such events must be supportive, inclusive, and safe environments for all participants. We believe that all participants are to be treated with dignity and respect. Discrimination and harassment cannot be tolerated. We are committed to ensuring that the conference « Homogeneous Spaces, Diophantine Approximation and Stationary Measures. » follows these principles. For more information on the Statement of Inclusiveness, see this dedicated web page http://www.math.toronto.edu/~rafi/statement/index.html. »
Mini-Cours :

Dense subgroups in simple groups
In this series of lectures, we will focus on simple Lie groups, their dense subgroups and the convolution powers of their measures. In particular, we will dicuss the following two questions.
Let G be a Lie group. Is every Borel measurable subgroup of G with maximal Hausdorff dimension equal to the group G?
Is the convolution of sufficiently many compactly supported continuous functions on G always continuously differentiable?
Even though the answer to these questions is no when G is abelian, the answer is yes when G is simple. This is a joint work with N. de Saxce. First, I will explain the history of these two questions and their interaction. Then, I will relate these questions to spectral gap properties. Finally, I will discuss these spectral gap properties. »

Dynamics on homogeneous spaces and Diophantine approximation.
I will discuss approaches to several problems concerning values of linear and quadratic forms using the ergodic theory of group actions on the space of unimodular lattices, and more generally, on homogeneous spaces of semisimple Lie groups.

Dynamics on quotients of SL(2,C) by discrete subgroups.
We will discuss old and recent results on topological and measurable dynamics of diagonal and unipotent flows on frame bundles and unit tangent bundles over hyperbolic manifolds. The first lectures will be a good introduction to the subject for young researchers.
Conférenciers
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Variance estimates on spaces of lattices​

Exponents of Diophantine approximation

Approximation diophantienne sur les variétés

Shrinking targets on homogeneous spaces and improving Dirichlet’s Theorem

Counting and equidistribution of integral representations by quadratic norm forms in positive characteristic​

Dynamical approaches to automorphic functions and resonances, and
reduction theories for indefinite quadratic forms

  • Uri Shapira (Technion, Israel Institute of Technology)

Generalizing Benoist-Quint to homogeneous spaces of non-lattice type

Random walks on homogeneous spaces and diophantine approximation on fractals