Lie groupoids of symmetries and geometric structures on manifolds

• Grant number: 24/22841-1
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: April 01, 2025
• End date: March 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Marcos Martins Alexandrino da Silva
• Grantee: Luca Accornero
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 22/16097-2 - Modern methods in differential geometry and geometric analysis, AP.TEM

Abstract

The overall goal of this project is to investigate geometric structures transverse to foliations by using the associated groupoids of symmetries and their (higher) geometry.The idea that the set of symmetries of a geometric structure carries a lot of information on the structure itself appears ubiquitously in geometry and can be traced back to the seminal work of Klein and Lie. Studying symmetries via groupoids and higher structures allows to discover new phenomena, and to better understand already known ones. Moreover, many geometric structures on manifolds exhibits a lack of homogeneity that results in a set of symmetries naturally modeled by a Lie groupoid rather than a Lie group; in such examples, higher methods become a necessary tool.The goal of this project is to work along these lines to address three classes of problems concerning geometric structures, both in themselves and as transverse structures to (singular) foliations.(i) The construction of (cohomological) invariants -- having in mind Bott's vanishing theorem and the theory of characteristic classes of foliations.(ii) Deformation problems -- having as case-study the deformation of contact structure into foliations.(iii) Complexification problems -- motivated by several examples in the literature, such as the complexification of Riemannian metrics and symplectic structures.The unifying theme is the approach: the three problems can be approached by working on the associated Lie groupoid of symmetries, and making use of Lie theoretical tools. The study of symmetry in differential geometry, and its close relationship with Lie theory, is one of the central research themes at IME-USP. Indeed, the research plan connects extremely well with the research directions at IME-USP.