Differential geometry: Lie groupoids; Poisson geometry; deformations of geometric structures
University of Coimbra
Institution: Utrecht University
Year: 2016
My research focuses mainly on differential geometric approaches to deformation theory and to the study of singular (i.e., not smooth) spaces modelled via Lie groupoids, called differentiable stacks (for example, quotients by group actions). Lie groupoids provide an unified framework for the study of several geometric structures, including Lie groups, Lie group actions on manifolds, submersions, foliations, Poisson structures, etc.
Just as Lie groups model smooth symmetries of a single object, Lie groupoids describe smooth symmetries of several objects at once. A lot of the theory of Lie groupoids is inspired by Lie theory. But parts are closer to other areas, such as foliation theory, and symplectic geometry. Algebraic geometry can also be a big source of examples and techniques. For example for deformation theory of Lie groupoids.
A central problem in geometry is that of understanding the behaviour of geometric structures under deformations. Typically, the final goal is to describe the moduli space of such structures up to an appropriate equivalence relation. Some objects are rigid, so they cannot be deformed. Others can be deformed and we are interested to know in which ways that happens.
Each class of geometric structures comes with its deformation theory, generally including two parts: an algebraic part (a cohomology theory that controls such deformations), and analytic methods, including the implicit function theorem for Banach spaces, and generalizations of it, to pass from algebraic to geometric results - for example, deciding whether an object is rigid or not.
In studying deformations of Lie groupoids, I am interested in the relations to theories of other geometric and algebraic objects, such as Lie algebras, Poisson manifolds, and complex manifolds.
Some topics of ongoing projects that a potential student could work on are:
- Understanding better the space of Lie groupoids, either by algebraic (cohomological) or analytical techniques (Riemannian centre of mass; inverse function theorems);
- Deformation theory of singular spaces described by groupoids (differentiable stacks), perhaps with a compatible structure (symplectic, complex);
- Formal deformation theory of (symplectic) groupoids, and relations with quantization;
- Geometry of compact differentiable stacks (vector fields, dynamics, "submanifolds"...).
