Geometric Structures on Lie Groupoids and their Applications

群通常以某一给定对象的对称性出现。群组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。李群胚的概念，以及它的无穷小对应的李代数体的概念，在数学的各个分支及其应用中发挥着越来越重要的作用。例如，人们现在可以在高能物理的场论中的动力学公式中找到它们，在生物学中的种群动力学的各种模型中找到它们，或者在博弈论中的进化动力学中找到它们。在过去的二十年里，李群胚和李代数体理论经历了几个激动人心的发展，本项目的目的是进一步发展该理论的基本方面和重要的新应用。该项目包括与在欧洲和南美工作的该领域的各种研究人员的合作，以及对3000名博士生的培训。国际和平研究所将参与在外地组织几次国际会议、讲习班和暑期课程，并在UIUC每周举办一次研讨会。该项目分为四项主要任务。第一个任务继续对紧致型Poisson流形的PI研究，紧致型Poisson流形是Poisson几何中的中心对象，扮演着类似于李理论中紧致李代数的角色。第二个任务是致力于无多重数纤颤，也就是。非对易可积系统，并开启了对它的一般奇点的研究。第三个任务涉及严格(非形式)形变量子化的研究，目的是建立第一个已知的阻碍Poisson流形严格形变量子化存在的障碍。第四个任务涉及Cartan在几何中的许多分类问题下的实现问题，即那些可以用(有限维)G-结构族来表示的问题。它的目的是建立基于李群胚和李代数体的此类实现问题的理论，并给出分类问题的完整解决方案。所有这些任务都在很大程度上依赖于李群胚技术，其中许多技术还有待开发。这些想法中的一些是对之前由PI和合作者开发的结果和技术的延伸和扩展。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Groups typically arise as the symmetries of some given object. The concept of a groupoid allows for more general symmetries, acting on a collection of objects rather than just a single one. The concept of a Lie groupoid, as well as its infinitesimal counterpart, the notion of a Lie algebroid, have played an increasing important role in various branches of mathematics, as well as its applications. For example, one can now find them in the formulation of dynamics in field theory in High Energy Physics, in various models for populations dynamics in Biology, or in evolutionary dynamics in Game Theory. Over the last two decades, the theory of Lie groupoids and Lie algebroids has undergone several exciting developments and this project aims to develop further both foundational aspects and significant new applications of the theory. This project includes collaborations with various researchers in the field working in Europe and South America and the training of 3 PhD students. The PI will be involved in the organization of several international meetings, workshops and summer courses in the field, as well as a weekly seminar at UIUC.The project is organized into 4 main tasks. The first task continues the PI study of Poisson manifolds of compact type, which are central objects in Poisson Geometry, playing a role analogous to compact Lie algebras in Lie Theory. The second task is dedicated to multiplicity free fibrations, a.k.a. non-commutative integrable systems, and initiates the study of its generic singularities. The third task concerns the study of strict (non-formal) deformation quantization and aims to establish the first known obstructions to the existence of strict deformation quantizations of a Poisson manifold. The fourth task concerns Cartan’s realization problem underlying many classification problems in geometry, namely the ones that can be formulated in terms of (finite dimensional) families of G-structures. It aims to establish a theory of such realization problems, based on Lie groupoids and Lie algebroids, as well as to give a complete solution to the classification problem. All these tasks rely heavily on Lie groupoid techniques, many of which have yet to be developed. Some of these ideas extend and expand results and techniques previously developed by the PI and collaborators.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.