Higher Structures and Groupoids in Noncommutative Geometry

这个项目涉及非对易几何的问题。非交换几何的思想是使用非交换代数来研究几何，非交换代数是具有加法和乘法的数学对象，但是乘法不是交换的：xy不必等于yx。 该项目的目的是围绕高级结构和非对易几何，研究这些领域中由物理学激发的数学问题。更具体地说，它是由量子力学、量子场论、弦论、变形量子化、微分几何、非对易几何和算子代数的思想组合而成的。拟议项目的跨学科性质促进了这些领域之间的进一步互动。PI继续通过在会议和研讨会上发言以及组织讲习班来传播其研究成果。 该项目的一些部分是与来自中国和欧洲的合作者共同完成的。因此，该项目为PI提供了一个与年轻科学家合作的绝佳机会，并与其他国家的同事交流思想，促进合作。非对易几何的想法是通过非对易流形上的函数代数来研究几何。在这样一个非交换流形上，相关的对象不再是空间中的点，而是一个非交换结合代数，或微分分次交换代数。许多非交换空间可以从广群得到。PI建议继续研究形变量子化，微分分次流形（DG流形），Chern-Connes特征和离域扭曲等变上同调。这些问题包括探索同伦李代数结构出现在连接与推广的Atiyah类的各种情况下，并制定了德拉姆模型的等变扭曲K理论使用李群胚和可微堆栈的理论。

This project involves problems in noncommutative geometry. The idea of noncommutative geometry is to study geometry using noncommutative algebras, which are mathematical objects that have addition and multiplication, however the multiplication is not commutative: xy does not have to equal yx. The purpose of the project, which is centered around higher structures and noncommutative geometry, is to investigate mathematical problems in these fields motivated by physics. More specifically, it is motivated by a combination of ideas from quantum mechanics, quantum field theory, string theory, deformation quantization, differential geometry, noncommutative geometry, and operator algebras. The interdisciplinary nature of the proposed project promotes further interaction between these fields. The PI continues to disseminate his research by speaking at conferences and seminars and organizing workshops. Some parts of the project are joint work with collaborators from China and Europe. The project thus provides an excellent opportunity for the PI to work with young scientists and to exchange ideas with colleagues from other countries and to promote collaborations.The idea of noncommutative geometry is to study geometry via algebras of functions on noncommutative manifolds. On such a noncommutative manifold, the relevant objects are no longer points in a space, but rather a noncommutative associative algebra, or a differential graded commutative algebra. Many noncommutative spaces can be obtained from groupoids. The PI proposes to continue the study of deformation quantization, differential graded manifolds (DG manifolds), Chern--Connes character, and delocalized twisted equivariant cohomology. The problems include exploring the homotopy Lie algebra structures appearing in connection with a generalization of the Atiyah class to various contexts, and developing a de Rham model of equivariant twisted K-theory using the theory of Lie groupoids and differentiable stacks.