Global Problems in Poisson Geometry and Related Structures

本计画旨在研究流形上泊松结构的性质，以及相关的几何结构，如狄拉克结构、李代数胚或广义复结构。该项目主要集中在全球方面，从叶状理论，等变几何，积分仿射几何和辛几何的思想和技术绘图。这些思想，加上李群胚理论在过去10年中发展的最新结果和技术，将导致新的方法来攻击泊松几何中一些长期存在的基本问题，如正则泊松结构的存在性，“紧型”泊松流形的分类，或超越线性化的叶子周围的正规形的存在性。该项目还旨在通过提出与外微分系统理论和可积系统理论的新相互作用来超越泊松几何的当前边界。泊松几何位于数学物理和几何的交叉点上。它起源于经典力学的数学表述，作为量子力学的半经典极限。在过去的20年里，由于与数学和数学物理的大量领域的联系，该领域发展迅速，包括微分几何和李理论，量子化，非对易几何，表示论和量子群，几何力学和可积系统。这个项目分享了泊松几何的这种味道，不仅旨在更深入地理解泊松括号的几何性质，而且还旨在开发其他数学领域的新应用。

This project aims to study properties of Poisson structures on manifolds as well as of related geometric structures such as Dirac structures, Lie algebroids or generalized complex structures. The project focus primarily on global aspects, drawing on ideas and techniques from Foliation Theory, Equivariant Geometry, Integral Affine Geometry and Symplectic Geometry. These ideas, together with recent results and techniques in Lie groupoid theory developed in the last 10 years, will lead to new methods to attack some long standing fundamental problems in Poisson Geometry, such as the existence of regular Poisson structures, the classification of Poisson manifolds of "compact type", or the existence of normal forms around leaves that go beyond linearization. The project also aims at going beyond the current boundaries of Poisson Geometry by proposing new interactions with the theory of Exterior Differential Systems and with the theory of Integrable Systems.Poisson Geometry lies on the intersection of Mathematical Physics and Geometry. It originates in the mathematical formulation of classical mechanics as the semiclassical limit of quantum mechanics. The field developed rapidly in the last 20 years, stimulated by the connections with a large number of areas in mathematics and mathematical physics, including differential geometry and Lie theory, quantization, noncommutative geometry, representation theory and quantum groups, geometric mechanics and integrable systems. This project shares this flavor of Poisson geometry, aiming not only at a deeper understanding of geometric properties of Poisson brackets, but also at developing new applications in other fields of Mathematics.