Data: 15/03/2017
Hora: 15:05h
Local: C116

Palestrante: Roberto Rubio (IMPA)

Título: Higher analogues of symplectic, Poisson and Dirac structures 

Resumo: We will first review how symplectic and Poisson structures on a manifold M fit into Dirac structures (integrable Lagrangian subbundles of TM+T*M), and then introduce higher versions of symplectic and Poisson structures. Having a higher version of Dirac structures is more troublesome than expected. Indeed, whereas the graph of a Poisson structure is a Dirac structure, and hence Lagrangian, the graph of a higher-Poisson structure, lying on TM plus the k-th exterior power of T*M, is not necessarily Lagrangian. To fix this, we introduce the notion of a weak-Lagrangian subspace and define higher-Dirac structures as integrable weak-Lagrangian subbundles, so that higher-Poisson structures do correspond to higher-Dirac structures transversal to TM. Higher-Dirac structures are, in particular, Lie algebroids, so their projection to TM is an involutive distribution and hence induces a (singular) foliation on M. A precise linear algebraic description of weak-Lagrangian subspaces allows us to describe the geometry appearing on the induced foliation, which is a non-trivial higher version of the presymplectic foliation of a Dirac structure. This is joint work with H. Bursztyn and N. Martínez-Alba.