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Geometric Structures on Lie Groupoids and their Applications

李群形上的几何结构及其应用

基本信息

批准号：
2003223
负责人：
Rui Loja Fernandes
金额：
$ 26.91万
依托单位：
University of Illinois at Urbana-Champaign
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2003223&HistoricalAwards=false
关键词：
Geometric Structures Lie Groupoids their

项目摘要

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.

群通常作为某个给定对象的对称性而出现。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。李群胚的概念，以及它的无穷小对应物，李代数胚的概念，在数学的各个分支及其应用中扮演着越来越重要的角色。例如，我们现在可以在高能物理学的场论动力学公式中，在生物学的种群动力学的各种模型中，或者在博弈论的进化动力学中找到它们。在过去的二十年里，李群胚和李代数胚的理论经历了几个令人兴奋的发展，这个项目的目的是进一步发展这两个理论的基础方面和重要的新应用。该项目包括与在欧洲和南美洲工作的各种研究人员的合作，以及3名博士生的培训。PI将参与组织几次国际会议，研讨会和暑期课程，以及在UIUC每周一次的研讨会。该项目分为4个主要任务。第一个任务继续PI研究的泊松流形的紧凑型，这是中心对象的泊松几何，发挥类似的作用，李理论中的紧李代数。第二个任务是致力于多重自由纤维化，a.k.a.非交换可积系统，并开始研究其一般奇点。第三个任务涉及严格（非形式）变形量子化的研究，目的是建立第一个已知的障碍存在的严格变形量子化的泊松流形。第四项任务涉及Cartan的实现问题的基础上许多分类问题的几何，即那些可以制定的（有限维）家庭的G-结构。它的目的是建立这样的实现问题的理论，基于李群胚和李代数胚，以及给一个完整的解决方案的分类问题。所有这些任务在很大程度上依赖于李群胚技术，其中许多尚未开发。其中一些想法扩展和扩展了PI和合作者先前开发的结果和技术。该奖项反映了NSF的法定使命，并被认为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来支持。