Symplectic groupoids and quantization of Poisson manifolds

群体通常以与特定对象相关的对称形式出现。然而，类群的概念允许更一般的对称作用于对象的集合，而不仅仅是一个对象。本项目通过结合类群技术和半经典分析来解决微分几何和量化数学理论中重要的基本问题。该项目的意义在于，它有可能揭示几何的新基本方面，并解开围绕物理学中量子化本质的谜团，这对于理解我们宇宙的几何方面至关重要。该项目包括与来自欧洲和南美的研究人员合作。此外，该项目的首席研究员将参与组织国际会议、夏季课程和伊利诺斯大学厄巴纳-香槟分校的每周一次研讨会。该项目包括三个主要任务。第一项任务介绍了泊松流形的非形式化变形量化的一种新方法，其中星积具有由半经典傅立叶积分算子定义的核。这种方法有两个目的：通过辛群类群建立与可积性的联系，并证明大范围泊松流形的星积的存在性。第二个任务集中于PI对紧型泊松流形（PMCT）的研究，它是泊松几何中的中心对象，类似于李论中的紧李代数。目前的目标是在非正则情况下发展理论，研究pmct的哈密顿空间，扩展李群作用的经典结果。这包括研究辛环面束的哈密顿空间，它推广了辛环流形。第三项任务继续探索几何结构分类问题的方法，使用Cartan的实现和由PI及其合作者首创的李群技术。PI将扩展先前关于有限维族的工作，以包括无限维的情况。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

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.