Singular Foliations: Desingularization and the Baum-Connes Conjecture
Final Report Abstract
The Baum-Connes conjecture in the context of singular foliations. The most common viewpoint on mathematics is that it concerns numbers (Number Theory) and solving differential equations (Analysis). Remarkably, results in both fields arise from the understanding of the symmetries of the problems involved: Number theory uses representations; on the other hand, the work of Sophus Lie in the 19th century established the groups with a prominent role in finding solutions. This brings in Geometry: due to Felix Klein, Geometry is the study of symmetries. The puzzle is completed by the works of Poincaré and Hilbert, namely Topology and Algebra. A big problem with differential equations is what happens at collapsing points of a solution (singularities), e.g. x is meaningless at zero. In physics (e.g. relativity) singularities account for phenomena such as black holes and the big bang. A Geometric model is foliation theory, namely partitions of a space to subspaces of different dimension (leaves). The crux is understanding the highly pathological topology of the space of leaves. Androulidakis and Skandalis in 2006 gave a workable model for this space (holonomy groupoid). The Baum-Connes conjecture (BC) is the latest of a long sequence of important achievements relating Geometry with Topology and Analysis. One way to understand the statement of BC is that all the Analytic representations of the situation involved come from Geometry. Apart from the validity of BC in various cases, its properties imply extremely deep results such as the Novikov and the Kadison-Kaplansky conjectures. BC is known to hold in a great deal of geometric situations, and counterexamples were given in 2001 by Higson, V. Lafforgue and Skandalis for certain foliations without singularities. As a result of this project, we managed to formulate BC for a very large class of singular foliations, provide conditions for its validity and make specific calculations. Important applications of this work are expected to arise in representation theory. Almost regular Poisson manifolds: The zeroth step towards a hierarchy in Poisson Geometry. Poisson Geometry arises from the works of Jacobi, D. Poisson and S. Lie in the 19th century. It accommodates the study of classical mechanics. In the 1970’s A. Weinstein’s Splitting Theorem gave the normal form of a Poisson manifold: It is a partition to symplectic leaves, usually with singularities. Actually such symplectic (singular) foliations determine Poisson structures completely. The next step is to obtain a kind of hierarchy for Poisson structures reflecting the nature of the singularities involved. We propose that this is done in terms of Algebra, by studying the module F generated by the Hamiltonian vector fields. In particular, we focus on the “zeroth” step of this hierarchy. Important examples of such “almost regular” Poisson structures are log-symplectic manifolds and the Lie-Poisson structure associated with the Heisenberg group. We prove that they are all determined by the leaves of the symplectic foliation. Moreover, it shows that the holonomy groupoid H(F), introduced originally by C. Debord is a good desingularization. In fact, H(F) gives several examples of Poisson groupoids, filling a gap as far as the scarcity of examples as such are concerned. Moreover, we show that the topology of the holonomy groupoid really controls the integrability of such Poisson structures. But what happens with Poisson structures such that F is not projective? We derive a method to perform resolution of singularities in the spirit of Algebraic Geometry. In particular, this means that a hierarchy of Poisson structures should be given by the length of a resolution associated with F. Very recently this was worked out in the PhD thesis of S. Lavau. We show that his work suggests that the integration such resolutions provides a desingularization of Poisson structures by means of “higher structures”.
Publications
- A Baum-Connes conjecture for singular foliations
I. Androulidakis and G. Skandalis
- Chern-Simons, Wess-Zumino and other cocycles from Kashiwara-Vergne and associators
A. Alekseev, F. Naef, X. Xu, C. Zhu
- Iterated crossed products for groupoid fibrations
A. Buss and R. Meyer
- Large time limit and local L2 - index theorems for families. J. Noncommut. Geom. 9 621–664 (2015)
S. Azzali, S. Goette and T. Schick
(See online at https://doi.org/10.4171/JNCG/203) - Stefan-Sussmann singular foliations, singular subalgebroids, and their associated sheaves. Int. J. Geom. Methods Mod. Phys. 13 1641001 (2016
I. Androulidakis and M. Zambon
(See online at https://dx.doi.org/10.1142/S0219887816410012) - Almost regular Poisson manifolds and their holonomy groupoids. Sel. Math. New Ser. (2017)
I. Androulidakis and M. Zambon
(See online at https://doi.org/10.1007/s00029-017-0319-5)
