Symplectic groupoids and quantization of Poisson manifolds

组通常以与特定对象相关联的对称性出现。然而，广群的概念允许更一般的对称性作用于对象的集合，而不仅仅是单个对象。这个项目解决了微分几何和量子化的数学理论的重要性的基本问题，采用了广群技术和半经典分析相结合。该项目的意义在于它有可能揭示几何学的新的基本方面，并解开物理学中量子化性质的奥秘，这对于理解我们宇宙的几何方面至关重要。该项目涉及与来自欧洲和南美洲的研究人员的合作。此外，该项目的首席研究员将参与组织国际会议，夏季课程，并在伊利诺伊大学香槟分校每周一次的研讨会。该项目包括三个主要任务。第一个任务介绍了一种新的方法来非正式变形量子化的泊松流形，其中的星星产品的核定义的半经典傅立叶积分算子。这种方法有两个目标：通过辛群胚建立可积性的联系，并证明大范围泊松流形的星星积的存在性。第二个任务主要是PI对紧型泊松流形（PMCT）的研究，这是泊松几何中的中心对象，类似于李理论中的紧李代数。目前的目标是在非正则的情况下发展的理论和研究的哈密顿空间的PMCT，推广李群作用的经典结果。这包括研究辛环面丛的哈密顿空间，它推广了辛环面流形。第三个任务继续探索的方法来分类问题的几何结构使用嘉当的实现和采用李群胚技术，开创了PI和合作者。PI将扩展以前在有限维族方面的工作，以包括无限维的情况。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估的支持。

Groups typically emerge as symmetries associated with a specific object. However, the concept of a groupoid allows for more general symmetries that act on collections of objects rather than just a single one. This project tackles fundamental questions of importance in differential geometry and the mathematical theory of quantization by employing a combination of groupoid techniques and semi-classical analysis. The project's significance lies in its potential to uncover new fundamental aspects in geometry and unravel mysteries surrounding the nature of quantization in physics, which are crucial for understanding the geometric aspects of our universe. The project involves collaborations with researchers from Europe and South America. Additionally, the project's lead investigator will be involved in the organization of international meetings, summer courses, and a weekly seminar at the University of Illinois, Urbana-Champaign.The project consists of three main tasks. The first task introduces a novel approach to non-formal deformation quantization of Poisson manifolds, where the star products have kernels defined by semi-classical Fourier integral operators. This approach has two objectives: establishing a connection to integrability through symplectic groupoids and proving the existence of star products for a wide range of Poisson manifolds. The second task focuses on the PI's research on Poisson manifolds of compact type (PMCT), which are central objects in Poisson geometry analogous to compact Lie algebras in Lie Theory. The current goals are to develop the theory in the non-regular case and study Hamiltonian spaces of PMCTs, extending classical results for Lie group actions. This includes investigating Hamiltonian spaces of symplectic torus bundles, which generalize symplectic toric manifolds. The third task continues the exploration of an approach to classification problems of geometric structures using Cartan's realizations and employing Lie groupoid techniques, pioneered by the PI and collaborators. The PI will expand previous work on finite-dimensional families to include the infinite-dimensional case.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.