(Very!) Singular Foliations: Treatment and Use
Foliations appear in the study of Dynamical Systems. Some classical Geometric examples are orbits of Lie group actions and the symplectic foliation of a Poisson manifold. In most cases these foliations are singular. Important examples of singular foliations also arise in Representation Theory, e.g. the Bruhat decomposition of a Grassmannian. And in Analysis: Hoermander showed that the propagation of singularities follows the integral curves of the Hamiltonian vector field associated with the principal symbol. In general, singular foliations are much more common than the regular ones.
I'm concerned with (very) singular foliations. My work focuses on understanding the topology of the associated leaf space, which is usually quite pathological. I use Noncommutative Geometry (NCG) for this purpose.
NCG replaces topological spaces with suitable operator algebras. This way, the inherent topological pathology is reflected to the noncommutativity of the convolution product. The associated K-theory then accommodates the topological invariants involved, such as characteristic classes and index theory. Also the spectral theory of the operators involved. By and large, my work carries out this program for the leaf space of a singular foliation.
In order to build the operator algebra which replaces a topological space, we need to understand its external symmetries. The geometric objects which describe such symmetries are groupoids. They give rise to algebras of pseudodifferential operators. In the case of foliations, holonomy provides the groupoid which models the leaf space. For singular foliations in particular, the notion of bisubmersion provides a very stable way to keep track of holonomy and eventually build the associated groupoid. Groupoids also play a crucial role in Mathematical Physics, where one can find lots of singularities.
The Baum-Connes conjecture establishes a strong relation of all this with Representation theory. One question is to what extend it is possible to use the Noncommutative Geometry of singular foliations in order to understand better certain aspects of Representation Theory. Also, somewhere in the intersection of Representation Theory with sub-Riemannian Geometry, it seems that bisubmersions can also be used to build (hard!) Analysis on objects more general than singular foliations. Such objects are singular foliations endowed with a (Lie) filtration, and generalised smooth distributions.