Titles and abstracts

Barbara Schapira (Rennes): Unipotent flows on infinite volume hyperbolic manifolds

Unipotent flows of hyperbolic manifolds $\Gamma\backslash \mathbb{H}^n$ of dimension at least 3 can be seen as a generalization of horocyclic flow of a hyperbolic surface. In a joined work with François Maucourant, we prove that when $\Gamma$ is Zariski dense, these unipotent flows are topologically transitive on their nonwandering set. Moreover, the natural invariant measure, the so-called Burger-Roblin measure, is U-ergodic, as soon as the critical exponent is large enough and the geodesic flow admits a finite measure maximizing entropy. It is a generalization of a result of Mohammadi-Oh in the convex-cocompact case.

Jens Marklof (Bristol): Spherical averages in the space of marked lattices

A marked lattice is a $d$-dimensional Euclidean lattice, where each lattice point is assigned a mark via a given random field on $\mathbb{Z}^d$. We prove that, if the field is strongly mixing with a faster-than-logarithmic rate, then for every given lattice and almost every marking, large spheres become equidistributed in the space of marked lattices. A key aspect of our study is that the space of marked lattices is not a homogeneous space, but rather a non-trivial fiber bundle over such a space. As an application, we prove that the free path length in a crystal with random defects has a limiting distribution in the Boltzmann-Grad limit. This is joint work with Ilya Vinogradov (Princeton).

Amie Wilkinson (Chicago):Geodesics on incomplete, negatively curved cusped surfaces

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