Homotopy Algebras in Noncommutative Geometry

这个项目是关于非对易几何问题的研究。非对易几何的思想是使用非对易代数来研究几何，非对易代数是具有加法和乘法运算的数学对象；然而，乘法可能不是对易的：xy不必等于yx。该项目的目的是研究一组由物理引发的数学问题。更具体地说，这一动机来自量子力学、弦理论以及代数和几何等经典数学领域的观点的结合。这项研究的跨学科性质促进了这些领域之间的进一步互动。该项目为研究人员提供了与年轻科学家合作并与其他国家的同事交流思想以促进科学合作的极好机会。在非对易几何中，人们通过非对易流形上的函数代数来研究几何。在这样的非交换流形上，相关的对象不再是空间中的点，而是一个非交换结合代数，或一个微分分次交换代数。作为交换代数的变形，可以得到一类重要的非交换流形。形变量子化理论介于经典力学和量子力学之间。这两种理论的数学结构截然不同。量子化，粗略地说，是对量子现象的研究和预测，这些现象通常由非对易结合代数描述，来自其潜在的经典对应的几何。本课题将集中研究非对易几何中的同伦代数结构，利用形变量子化和李群胚和李代数体理论的工具。这些问题包括探索Duflo和Todd型类，建立Kontsevich-Duflo型定理，以及在李代数体的一般框架下研究Tsygan非对易演算。

This project concerns the investigation of problems in noncommutative geometry. The idea of noncommutative geometry is to study geometry using noncommutative algebras, which are mathematical objects that have operations of addition and multiplication; however, the multiplication may not be commutative: xy does not have to equal yx. The purpose of the project is to investigate a set of mathematical problems motivated by physics. More specifically, the motivation derives from a combination of ideas from quantum mechanics, string theory, and classical areas of mathematics such as algebra and geometry. The interdisciplinary nature of the research promotes further interaction between these fields. The project provides excellent opportunities for the investigator to work with young scientists and to exchange ideas with colleagues from other countries to promote scientific collaborations.In noncommutative geometry, one studies 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. An important class of noncommutative manifolds can be obtained as deformations of commutative algebras. The theory of deformation quantization lies on the boundary between classical and quantum mechanics. The mathematical structures of the two theories are very different. Quantization, roughly speaking, is the study and prediction of quantum phenomena, which are normally described by noncommutative associative algebras, from the geometry of their underlying classical counterparts. This project will focus on the study of homotopy algebra structures in noncommutative geometry using tools from deformation quantization and Lie groupoid and Lie algebroid theory. The problems include exploring the Duflo and Todd type class, establishing a Kontsevich-Duflo type theorem, and studying Tsygan noncommutative calculi in a general framework in terms of Lie algebroids.