Classification problems and moduli spaces of geometric structures via Lie Theory

• Grant number: 22/16310-8
• Support Opportunities: Regular Research Grants
• Start date: April 01, 2023
• End date: March 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Ivan Struchiner
• Grantee: Ivan Struchiner
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

In the field of differential geometry it is common that one takes interest in geometric structures which satisfy a natural (invariant under diffeomorphisms) differential equation. For a fixed geometric structure and a differential equation defined on the space of such structures, the main objective is to describe the moduli space of the structures under investigation, as well as to classify all manifolds which admit such structures.In this project we propose questions related classification problems of geometric structures subjected to a differential equation of finite type. Such problems, known as Cartan's Realization Problem, were studied by the proponent of this grant in the context of G-structures with connections in.Here we propose several directions in which it is possible to extend and/or deepen the results obtained previously. On the one hand, we propose to study classification problems for geometric structures which are more general than G-structures with connections. On the other hand, we propose to study the particular case where the classification problem is controlled by a Poisson structure. (AU)