"Noncommutative Geometry for Sigular Foliations"

FP7-PEOPLE-2011-CIG, Grant No PCI09-GA-2011-290823

(This research project is now completed.)

Objectives

The Baum-Connes conjecture (BC) is a far-reaching generalization of the Atiyah-Singer index theorem. It uses index theory to establish a link (assembly map) between the K-theory of convolution algebras of geometric origin (analytical side) with homological invariants of the geometric situation involved (topological side), and conjectures that it is an equivalence. Counterexamples to BC were given by Higson, V. Lafforgue and Skandalis. Even so, the injectivity and surjectivity of the assembly map provide far deeper information for the geometric situation involved, as they imply the validity of the Novikov and the Kadison-Kaplansky conjectures. The purpose of this research proposal is to formulate and study BC for singular foliations. They are the most common ones (e.g. in dynamical systems, control theory, math. physics, Poisson geometry, etc.), and the least well-studied. Particularly, it will show that the injectivity/surjectivity of the assembly map, depends only on its behaviour on each stratum of equal-dimensional leaves.

The project will build on important recent results on singular foliations by I. Androulidakis and G. Skandalis. Extending work of Connes on non-Hausdorff groupoids, they attached to a singular foliation the holonomy groupoid and the convolution algebra(s). They also defined the associated longitudinal pseudodifferential calculus and the associated analytic index map. Work of Baum, Connes, Higson, as well as Le Gall and Tu shows that these are all the necessary ingredients to define the assembly map.

Publications

[1] I. Androulidakis and M. Zambon. Holonomy transformations for singular foliations. Advances in Mathematics 256 (2014) 348-397. (arXiv:1205.6008)

[2] I. Androulidakis and M. Zambon. Smoothness of holonomy covers for singular foliations and essential isotropy.
Math. Z. DOI 10.1007/s00209-013-1166-5. (arXiv:1111.1327)

[3] I. Androulidakis. Laplacians and spectrum for singular foliations. Chin. Ann. Math. Ser. B 35 (2014), no. 5, 679–690.