Geometric structures and where to find them

• Applicant: Dr. Francesco Cattafi
• Subject Area: Mathematics
• Term: from 2021 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 460397678

Mathematicians have always been fascinated with symmetries, shapes and patterns, and for centuries have studied different ways to describe them. The modern geometric reformulation makes use of the concept of manifold, i.e. a space which is locally modelled on a flat space. Examples include the 3-dimensional space and the 4-dimensional space-time (which are globally flat), as well as curved spaces, such as the surface of a sphere or the universe of general relativity: those are only locally flat, because the area around each point can be approximated by a flat space.Manifolds can also be endowed with extra objects called geometric structures, which can be applied to many areas of mathematics and physics. These are tools to performspecific tasks: for instance, a Riemannian metric measures distances between points, while a foliation partitions the manifold in smaller pieces.Many geometric structures can be defined using the symmetries of a given space. However, most contemporary research focusses on specific examples or is limited to the case when the symmetries form a group. Despite attempts in the past to treat various geometric structures at once, the current situation is not satisfactory yet. In order to develop a more complete theory, one should allow for local symmetries as well, described by more general versions of groups: namely, pseudogroups. The associated theory of geometric structures, called Γ-structures, is quite old: it encompasses almost every known example, but is still seriously underdeveloped. The main reason is that the proper tools to study it were not available until a few years ago.In order to develop a more complete theory, one should allow for local symmetries as well, described by more general versions of groups: namely, pseudogroups. The associated theory of geometric structures, called Γ-structures, is quite old: it encompasses almost every known example, but is still seriously underdeveloped. The main reason is that the proper tools to study it were not available until a few years ago.The aim of my proposal is to pursue this development, by proving several results on the integrability and classification of Γ-structures. An important aspect of this project is the use of recent and powerful techniques involving Lie groupoids, multiplicative forms and Morita equivalences, as well the development of new ones. I plan as well to use this approach to shed light on other theories of geometric structures, such as G-structures and Cartan geometries.

• DFG Programme: WBP Position