Geometric structures via Lie Theory

• Grant number: 15/50472-1
• Support Opportunities: Regular Research Grants
• Duration: August 01, 2016 - July 31, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Convênio/Acordo: University of Illinois
• Mobility Program: SPRINT - Projetos de pesquisa - Mobilidade
• Principal Investigator: Ivan Struchiner
• Grantee: Ivan Struchiner
• Principal researcher abroad: Rui Loja Fernandes
• Institution abroad: University of Illinois at Chicago (UIC), United States
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 15/22059-2 - Geometry and topology via Lie Theory, AP.R

Abstract

Lie groups are well known to play a central role in modern differential geometry. In fact, since the seminal work of Felix Klein, the very definition of a geometric structure has become intertwined to its symmetry Lie group. However, there are many geometric structures which cannot be described using Lie groups. This may occur for several reasons. Some structures do not have a preferred (canonical) local model (e.g., Poisson structures, Dirac structures; generalized complex structures, etc...). Even for classical geometric structures (G-structures), when dealing with equivalence problems, and classification (or moduli) problems, the symmetries involved are described by more general objects than Lie groups. This project deals with geometric structures determined by Lie groupoids. Moreover, it follows the paradigm introduced by Felix Klein in his Erlangen Program, and takes the point of view that the geometry and Lie theory of Lie groupoids determines and is determined by the geometric structures it induces. In it, we explore classification and integrability problems in classical differential geometry by making use of the Lie theory of the Lie algebroids/groupoids which describe their symmetries. At the same time, we treat geometric and topological aspects of Lie groupoids themselves, and also of their associated fiber bundles. For example, we deal with Chern-Weil theory for principal bundles with a structure Lie groupoid. (AU)