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Geometric structures via Lie Theory

Grant number: 15/50472-1
Support type:Regular Research Grants
Duration: August 01, 2016 - July 31, 2018
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Cooperation agreement: University of Illinois
Mobility Program: SPRINT - Projetos de pesquisa - Mobilidade
Principal researcher:Ivan Struchiner
Grantee:Ivan Struchiner
Principal researcher abroad: Rui Loja Fernandes
Institution abroad: University of Illinois at Chicago (UIC), United States
Home 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)

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