Mathis Alleysson : Quid of index theory?
Past development of index theory led to the introduction of index morphisms in K-theory. Unfortunately, these indices are not induced by the classical K-theory functor, as they go in the "wrong-way". In this presentation, our goal is to introduce push-forwards in K-theory, a functor solving this "wrong-way" issue. To do so, we will have to introduce some notions and examples of Lie groupoids, such as the deformation to normal cone. We might also have time to discuss the functoriality of push-forwards and how it can be applied to rewriting a proof of one of Atiyah-Singer's famous theorems.
Guilherme Sobreira : Quid of Anosov surfaces?
Anosov surfaces can be seen as a dynamical generalization of surfaces of negative curvature. It turns out that these surfaces display a lot of rigidity phenomena: invariants defined using the metric, like the spectrum of the Laplacian or lengths of geodesics, may often determine the metric. This is the case of the marked length spectrum, which we will discuss in this seminar. To understand why this invariant determines the underlying Anosov metric, we will come across a beautiful interplay between the complex structure of our surface and the dynamical/analytical behavior of its geodesic flow.
Duez Gabriel : Quid of Cartan subalgebras in C*-algebras?
Cartan subalgebras are particular maximal abelian subalgebras which exist in various theories. The most basic example is the subalgebra of diagonal matrices in a matrix algebra. For operator algebras, Cartan subalgebras were first defined in the von Neumann case and were proved to be equivalent to some particular measured equivalence relations. By analysing groupoid C*-algebras, Renault defined in 2008 Cartan subalgebras in C*-algebras and proved that they are equivalent to some groupoid structures. We propose to (re)discover this fundamental theorem, a bridge between dynamical systems and operator algebras.
