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Mini-Course
by
David Martinez Torres

"Poisson geometry from a symplectic perspective"

第１, ２回は中央大学，第３回は東工大で行われますのでご注意ください．

Abstract:   Poisson structures can be seen as generalizations of symplectic structures. This generalization is so flexible, that virtually no tool from symplectic geometry can be pushed to the Poisson setting. In this series of talks we aim at discussing up to which extent it is possible to select a class of Poisson structures for which some sort of `symplectic approach’ is possible.
      In more detail, we shall start with a brief introduction to Poisson geometry, stressing the many different approaches to the subject (classical mechanics, foliation theory, symplectic geometry). Next we shall describe a few fundamental problems in Poisson geometry and the difficulties to tackle them for arbitrary Poisson structures.
      We continue introducing the class on integrable Poisson manifolds: roughly, these are Poisson manifolds dominated by symplectic manifolds (complete symplectic realizations). Equivalently, a Poisson structure defines a Lie algebroid structure (an infinite dimensional Lie algebra of `geometric nature'), and, integrable Poisson manifolds, they are those Poisson manifolds for which their associated Lie algebroid integrates into a Lie groupoid (the kind of structure which formalizes `partial symmetries').
      We shall expend some time discussing how to recognize integrable Poisson manifolds (more generally, integrable algebroids), and, interestingly enough, we shall see how the integrability of a Poisson structure (of a general Lie algebroid) amounts to the smoothness of the leaf space of a foliation (on a Banach manifold).
      After describing up to which extend the integrability assumption exerts control on the Poisson structure, we will see how imposing additional `compact type conditions' on symplectic integrations leads to a class of (rather rigid) Poisson manifolds with, to some extent, are a simultaneous generalization of compact symplectic manifolds, compact Lie algebras and compact/proper foliations.

主催者からのコメント：
David Martinez Torres 氏に上記の表題で6コマのミニコースをお願いしています。 古典力学も含めて典型的な例に触れ、余/随伴軌道、symplectic 簡約，Kähler 構造、Lie Poisson 構造、Lie groupoid, Integrability などについて解説していただくことになると思います。

連絡先：三松　佳彦
TEL:03-3817-1749
E-MAIL:yoshiATmath.chuo-u.ac.jp