Geometric structures and where to find them
Final Report Abstract
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 modeled 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 perform specific 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. The standard mathematical concept to model (continuous) symmetries is that of a (Lie) group. However, in order to develop a more complete theory of geometric structures, one should allow for local symmetries as well, described by more general versions of groups, namely (Lie) groupoids. The current state of the art remains unsatisfactory; despite significant advancements in the theory of Lie groupoids over the last 30 years, their applications to geometric structures are still underdeveloped. In this project, I employed recent and powerful techniques, notably multiplicative structures and Morita equivalences, to address a number of challenging problems. These encompass specific geometric structures, such as foliations, as well as more general frameworks capable of treating multiple structures simultaneously, for instance, Cartan geometries and G-structures. The outcomes of my investigations have yielded promising results that are interesting on their own but also lay the groundwork for future research projects.
Publications
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A groupoid approach to transitive differential geometry
Francesco Cattafi & L. Accornero
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Pseudogroups of symmetries and Morita equivalences
Francesco Cattafi & L. Accornero
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Poisson manifold. WikiJournal of Science, 7(1), X.
Cattafi, Francesco
