This project will investigate group structures on differentiable stacks and in bimodule categories of algebras, concentrating on the example of Poisson and quantum tori. The project will also study complex Lie algebroids to gain insight into the relative index problem for Cauchy-Riemann structures on contact manifolds, and to construct holomorphic objects associated with generalized complex manifolds. Finally, applications of groupoids will be made to general relativity and other areas where groups are not flexible enough to encode the natural symmetries of a system.
In classical geometry, the notion of "space" is commonly identified with that of smooth manifold or some other kind of set with additional structure. But the study of global and local quotient objects, sometimes arising from concrete problems in mathematics, physics, and engineering, often leads beyond the world of smooth manifolds, and even beyond the world of sets, to objects known as (differentiable) stacks. In this new world, and the dual world of algebra, where geometric objects may be identified with algebras of functions, stacks become objects in a new category in which the isomorphisms are what is generally known as Morita equivalences. The project is devoted to understanding the corresponding notions of symmetry in these new worlds.
