In the classical theory, holomorphic quadratic differentials are tied to a wide range of objects, e.g. harmonic functions, minimal surfaces and Teichmüller space. We present a discretization of holomorphic quadratic differentials that preserves such a rich theory. We introduce discrete holomorphic quadratic differentials and discuss their connections to the surface theory and Teichmüller theory. On one hand, holomorphic quadratic differentials are related to discrete minimal surfaces via a Weierstrass representation. On the other hand, they arise from deformations of circle packings on surfaces with complex projective structures.

The Rauzy induction is the combinatorial counterpart of the Teichmuller flow. Its study is central in the proof of many of the dynamical results in Teichmuller dynamics such as the exponential mixing property, existence and unicity of an invariant measure of maximal entropy, ... After giving an introduction to this induction in the case of abelian and quadratic differentials, I will explain a generalization to multidimensional continued fractions, as well as a ergodicity criterion, and construction of invariant measures.  

The Atiyah-Singer index theorem expresses the index of the three fundamental geometric operators,
i.e. the signature operator, the spin-Dirac operator and the Riemann-Roch operator, in terms of
the L-genus, the A-roof genus and the Todd genus of the manifold. These genera have remarkable stability properties.
One can define higher genera by taking the fundamental group and its cohomology into account and deep results of Kasparov, Connes, Moscovici and many others express these numbers in terms of the K-homology class [D] associated to these operators, information that is then crucial in studying the stability properties of the higher genera.
In this talk I will begin by explaining this fascinating circle of ideas, together with its geometric applications, and then move on and give an answer to the following question: what of all this can be generalized to singular manifolds? As we shall see the answer depends heavily on microlocal analysis, at least for the first 2 examples.

The results that I will present are scattered in papers with Albin-Leichtnam-Mazzeo, Botvinnik- Rosenberg, Bei.

In joint work with Peter Hintz and András Vasy, we study the asymptotic behavior of linearized gravitational perturbations of Schwarzschild and slowly rotating Kerr black hole spacetimes. We show that solutions of the linearized Einstein equation decay at an inverse polynomial rate to a stationary solution (given by an infinitesimal variation of the mass and angular momentum of the black hole), plus a pure gauge term. In this talk, I will describe the geometric background and
the analytic setup of our result, including a discussion of gauge fixing. Our proof is based on a precise description of the resolvent of an associated wave equation on symmetric 2-tensors near zero energy.

According to a classical theorem of Steinhaus, if a subset E of d-dimensional Euclidean space
has positive Lebesgue measure, then its difference set, E-E, contains a neighbourhood of the
origin. Configuration set problems concern characterizing the sizes of collections of point
configurations that arise among the points of a subset of Euclidean (or more general) spaces; there are versions in discrete geometry (Erd\"os-Purdy type problems) for sets with a large number N of points, as well as in continuous geometry (Falconer type problems) for fractals with some lower bound on their dimension. An example of the latter is a result of Mattila and Sj\"olin that if the Hausdorff dimension of E is >(d+1)/2, then the set of distances between points of E contains an open interval. I will discuss a general approach to results of this type using Fourier integral operators, giving positive results for a wide variety of geometries, and also discuss examples where it fails.
This is joint work with Alex Iosevich and Krystal Taylor.

After reviewing the notion of geometrically finite hyperbolic metrics, I will explain how perturbing their conformal infinity yields new examples of complete Einstein metrics via the inverse function theorem. This is a joint work with Eric Bahuaud.

This will be a conference on the wide range of topics aimed to inspire future directions and new applications of microlocal methods.