For any symmetric space $X$ of noncompact type, its quotients by torsion-free discrete isometry groups $\Gamma$ are locally symmetric spaces. One problem is to understand the geometry and analysis, especially the spectral theory, and interaction between them of such spaces. Two classes of infinite groups $\Gamma$ have been extensively studied:
$(1) \Gamma$ is a lattice, and hence $\Gamma$ $\backslash$ $X$ has finite volume.
$(2) X$ is of rank $1$, for example, when $X$ is the real hyperbolic space, $\Gamma$ is geometrically finite and $\Gamma$ $\backslash$ $X$ has infinite volume.
When $\Gamma$ is a nonuniform lattice in case $(1)$ or any group in case $(2)$, compactification of $\Gamma$ $\backslash$ $X$ and its boundary play an important role in the geometric scattering theory of $\Gamma$ $\backslash$ $X$. When $X$ is of rank at least $2$, quotients of $X$ of finite volume have also been extensively studied. There has been a lot of recent interest and work to understand quotients $\Gamma$ $\backslash$ $X$ of infinite volume. For example, there are some generalizations of convex cocompact groups, but no generalizations yet of geometrically finite groups. They are related to the notion of thin groups. One naturally expects that these locally symmetric spaces should have real analytic compactifications with corners (with codimension equal to the rank), and their boundary should also be used to parametrize the continuous spectrum and to understand the geometrically scattering theory. These compactifications also provide a natural class of manifolds with corners. In this talk, I will describe some questions, open problems and results.[-]

Hyperbolic (Anosov or Axiom A) flows have discrete Ruelle spectrum. For contact Anosov flows, e.g. geodesic flows, where a smooth contact one form is preserved, the trapped set is a smooth symplectic manifold, normally hyperbolic, and M. Tsujii, S. Nonnenmacher and M. Zworski, have given an estimate for the asymptotic spectral gap, i.e. that appears in the limit of high frequencies in the flow direction. We will propose a different approach that may improve this estimate. This will be presented on a simple toy model, partially expanding maps. Work with Tobias Weich.[-]

Recently David Borthwick discovered through numerical calculations surprising chain structures in the resonance spectrum of certain Schottky surfaces. In this talk we will see that theses resonance chains have the same origin as the resonance chains in the classical and quantum mechanical spectrum of the three disk system and we will see that they are related to a clustering in the length spectrum. Finally the existence of these chains will be proven for three funneled Schottky surfaces in a certain geometrical limit in the Teichmüller space. Joint work with S. Barkhofen and F. Faure.[-]

In this lecture I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in 3+1 dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum.
I will discuss the geometry of these black holes, and then talk about the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted, Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. In this talk, I will emphasize geometric aspects of the stability problem, in particular showing that Kerr-de Sitter black holes with small angular momentum are stable in this sense.[-]

Over the last few years I developed (partly jointly with coauthors) dual 'slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for certain hyperbolic surfaces/orbifolds L \ H with cusps (both of finite and infinite area; arithmetic and non-arithmetic).
Both types of transfer operators arise from discretizations of the geodesic flow on L \ H. The eigenfunctions with eigenvalue 1 of slow transfer operators characterize Maass cusp forms. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. The Fredholm determinant of the fast transfer operators equals the Selberg zeta function, which yields that the zeros of the Selberg zeta function (among which are the resonances) are determined by the eigenfunctions with eigenvalue 1 of the fast transfer operators. It is a natural question how the eigenspaces of these two types of transfer operators are related to each other.[-]

Liouville CFT is a conformal field theory developped in the early 80s in physics, it describes random surfaces and more precisely random Riemannian metrics on surfaces. We will explain, using the Gaussian multiplicative chaos, how to associate to each surface $\Sigma$ with boundary an amplitude, which is an $L^2$ function on the space of fields on the boundary of $\Sigma$ (i.e. the Sobolev space $H^{-s}(\mathbb{S}^1)^b$ equipped with a Gaussian measure, if the boundary of $\Sigma$ has $b$ connected components), and then how these amplitudes compose under gluing of surfaces along their boundary (the so-called Segal axioms).
This allows us to give formulas for all partition and correlation functions of the Liouville CFT in terms of $3$ point correlation functions on the Riemann sphere (DOZZ formula) and the conformal blocks, which are holomorphic functions of the moduli of the space of Riemann surfaces with marked points. This gives a link between the probabilistic approach and the representation theory approach for CFTs, and a mathematical construction and resolution of an important non-rational conformal field theory.
This is joint work with A. Kupiainen, R. Rhodes and V. Vargas. [-]