Removing symplectic & Poisson singularities with Dirac structures
主 题: Removing symplectic & Poisson singularities with Dirac structures
报告人: Christian Blohmann (Max-Planck-Institut für Mathematik)
时 间: 2014-09-17 15:00-16:00
地 点: 理科一号楼 1418（主持人：刘张炬）
Dirac structures can be viewed as a method to study presymplectic and Poisson geometry in a unified way by considering not the 2-forms or bivectors but rather their graphs inside $TM \oplus T^*M$, e.g., viewing a presymplectic 2-form on $M$ as map $\omega: TM \to T^*M$. Beyond providing a common framework for known geometric structures, Dirac structures allow for generalized geometries that are of neither presymplectic nor Poisson type. An interesting class of examples is given by the following observation: The graph of the magnetic symplectic form of a magnetic monopole in 2 dimensions extends to a smooth Dirac structure over the singular locus of the monopole. I call such a singularity removable. The question whether a given singularity of a presymplectic or Poisson structure is removable in this sense is surprisingly subtle. For example, the singularity of a Dirac monopole in 3 dimensions is not removable. I will review the notion of Dirac structures, explain the example of the magnetic monopole in some detail, and report on recent results that provide a complete understanding of the question of removability and the structure of removable singularities. The key tool is a (new) local splitting theorem for Dirac structures.