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Short Lectures: An introduction to Lie pseudgroups and Geometric Structures  

 Speaker :

 Francesco Cattafi (Utrecht University)

 Organized by:

 Prof.  Ivan Struchiner                                           

 Date and place:
  •  November 26, 2019 – 14h00 às 16h00–room 165B
  • November 27, 2019 – 14h00 às 16h00–room 165B
  • November 28, 2019 – 14h00 às 16h00–room 165B
Description :In this minicourse I aim to present a general theory of geometric structures described by a Lie pseudogroup and to study the related integrability problem. Here is a rough plan of the three lectures.


 Lecture 1 - The problem: integrability of almost structures
I will start this lecture by presenting many classical examples of geometric structures on manifolds (e.g. symplectic, complex, contact, etc.) and explaining how they can all be described via a special smooth atlas. This leads to the general notion of a (Lie) pseudogroup $\Gamma$ and of a $\Gamma$-structures.
Inspired by the already existing theory of $G$-structures, we arrive to a definition of "almost" $\Gamma$-structures and to the problem of (formal) integrability. The key point of this new framework is to consider a principal bundle for a Liegroupoid action (and not just a Lie group one). In order to do that, I will recall first the basic notions of jet bundles, of Lie groupoids and Lie groupoid (principal) action. If time permits, I will also review the notion of Morita equivalence between Lie groupoids and show how/when "Morita equivalent pseudogroups" induce the same geometric structures.

Lecture 2 - The tools: principal Pfaffian bundles
Most of this lecture will be independent from and more abstract than the previous one. After recapping the definition of almost $\Gamma$-structure and the (formal) integrability problem, I will introduce the main concept of principal Pfaffian bundle. This is a principal groupoid bundle endowed with two differential forms satisfying several compatibility relations.
In order to interpret almost $\Gamma$-structures as an example, I review first the classical Cartan form on jet bundles. This leads to the natural definition of abstract almost $\Gamma$-structure and allows a better framework to study formal integrability.If time permits, I will discuss how to adapt the notion of Morita equivalence to the Pfaffian case and its consequence on (formal) integrability.

Lecture 3 - The solution: prolongations and intrinsic torsions
In this third lecture we will use the ingredients from lecture 2 in order to produce explicit obstructions to formal integrability. Inspired by the formal theory of PDEs, we will first develop a general theory of prolongations for principal Pfaffian bundles and then apply it to almost $\Gamma$-structures.
We will also consider some special cases, and in particular recover the classical obstructions to integrability for $G$-structures, i.e. the so-called intrinsic torsions.If time permits, I will sketch how to tackle the next natural problem, i.e. to find under which conditions formal integrability is sufficient for integrability.