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Stability of Lie brackets and their morphisms

Grant number: 13/16753-8
Support Opportunities:Regular Research Grants
Duration: October 01, 2013 - September 30, 2015
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


A central problem in differential geometry is to understand how geometric structures behave under deformations. Rigidity of a geometric structure means that every other geometric structure of the same kind which is sufficiently close to the original one is equivalent to it. This phenomenon has been observed in many different classical geometries, where it has furnished indispensible insight into the inner workings of these structures. This is perhaps most evident in the examples provided by Complex Geometry (Kodaira-Spencer), Symplectic Geometry (Darboux-Moser), Lie algebras (Nijenhuis-Richardson), Foliation Theory (Hamilton) and Singularity Theory (Mather).This fruitful approach to Geometry is a time-honored one, and yet until quite recently one lacked a sufficiently general framework under which that strategy could be systematically employed - a framework sufficiently broad to encompass all known instances of this method, and to guide the investigation of such deformation properties in Geometries where little or no rigidity is known.We propose one such general description in terms of Lie algebroids. Our aim is to unify the classical (and seemingly unrelated) theories mentioned above, while at the same time supplying tools to unearth rigidity phenomena in other fields of Geometry where known instances of such phenomena are scarce and usually derived in somewhat ad hoc fashion (e.g., Poisson Geometry).The research will be developed in three stages. The first is to determine the infinitesimal condition for stability of morphisms of Lie algebroids. In general, this translates into the vanishing of a cohomology group, and it is this cohomology theory that we will search for. The second step is to determine when infinitesimal stability implies actual stability. It is expected that this will be the hardest and most technical part of the research, where it will be necessary to apply some version of the Implicit Function Theorem (e.g., the Nash-Moser Theorem). The last stage of the research is to apply the results obtained to several examples. (AU)

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Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
CRAINIC, MARIUS; MESTRE, JOAO NUNO; STRUCHINER, IVAN. Deformations of Lie Groupoids. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, v. 2020, n. 21, p. 7662-7746, . (13/16753-8)
CRAINIC, MARIUS; SCHATZ, FLORIAN; STRUCHINER, IVAN. A survey on stability and rigidity results for Lie algebras. INDAGATIONES MATHEMATICAE-NEW SERIES, v. 25, n. 5, SI, p. 957-976, . (13/16753-8)
CRAINIC, MARIUS; SCHATZ, FLORIAN; STRUCHINER, IVAN. A survey on stability and rigidity results for Lie algebras. INDAGATIONES MATHEMATICAE-NEW SERIES, v. 25, n. 5, p. 20-pg., . (13/16753-8)

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