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RESEARCH SCHOOL

Homogeneous Spaces, Diophantine Approximation and Stationary Measures (CNRS Thematic School)
February 6 – 10, 2017
Homogeneous dynamics is the study of asymptotic properties of actions of subgroups of Lie groups on associated homogeneous spaces, including many geometric examples such as Anosov diffeomorphisms of the torus or geodesic flows on negatively curved manifolds. The theory has also deep connections with number theory and diophantine approximations, as shown by Margulis’ proof of Oppenheim conjecture and, more recently, the work of Einsiedler, Katok and Lindenstrauss on Littlewood conjecture.

This workshop will bring together researchers in the domain of homogeneous dynamics and its applications to arithmetic, and will feature the latest developments in the area as well as some mini-courses by some leading experts.
Scientific Committee & Organizing Committee

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-Courses

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.
Speakers 

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