About the project
The analysis of symmetry in Mathematics is performed through the notion of group. By generalizing this notion, not only we can broaden the applicability range of group-theoretic techniques, but we can also pick out the essential qualities of symmetry, retaining its core features while allowing for extra flexibility in its study. This leads us to the notion of groupoid, which is central in this project. We have particular interest in groupoids arising from topological dynamical systems.
A different approach is obtained by understanding the generalization procedure as a sort of deformation. This option takes us to the notion of quantum group, understood as a Hopf algebra of functions on a "noncommutative space".
Naturally, noncommutative geometry is an ideal framework to study these kind of generalized objects. In this project, we focus on the homological invariants we can associate to these mathematical entities.
We hope to develop a framework for computing and relating these invariants to one another, in analogy with the case of algebraic topology, where topological spaces are studied in a similar manner by constructing functors to algebraic structures.
Objectives
- Expand existing results on groupoid homology beyond the case of ample groupoids, and beyond the case of trivial isotropy (e.g., by means of Bredon-type homology).
- Study the relations between homology and K-theory of groupoid operator algebras.
- Analyze torsion phenomena, the concept of "proper action", and formulations of the Baum-Connes conjecture in the setting of quantum groups.
Background
The background material for this project includes, but is not limited to, the works below:
- Crainic, M.; Moerdijk, I. A homology theory for étale groupoids. J. Reine Angew. Math. 521, 25-46 (2000).
- Davis, J. F., Lück, W. Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory. K-Theory 15, No. 3, 201-252 (1998).
- Matui, H. Homology and topological full groups of étale groupoids on totally disconnected spaces. Proc. Lond. Math. Soc. (3) 104, No. 1, 27-56 (2012).
- Meyer, R., Nest, R. The Baum-Connes conjecture via localisation of categories. Topology 45, No. 2, 209-259 (2006).
Financing
This project is supported by MSCA, funded by the European Union.
Project number 101063362.