Homotopy Algebras in Noncommutative Geometry

非交换几何中的同伦代数

• 批准号: 1707545
• 负责人: Ping Xu
• 金额: $ 30万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1707545&HistoricalAwards=false
• 关键词: Homotopy; Algebras; Noncommutative; Geometry

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.

这个项目涉及非交换几何问题的研究。非交换几何的思想是使用非交换代数来学习几何，非交换代数是具有加法和乘法运算的数学对象；然而，乘法不一定是可交换的：xy不一定等于yx。该项目的目的是研究一组由物理学引起的数学问题。更具体地说，动机来源于量子力学、弦理论和经典数学领域（如代数和几何）的思想组合。该研究的跨学科性质促进了这些领域之间的进一步互动。该项目为研究者提供了与年轻科学家合作以及与来自其他国家的同事交流思想以促进科学合作的绝佳机会。在非交换几何中，人们通过非交换流形上的函数代数来研究几何。在这种非交换流形上，相关对象不再是空间中的点，而是一个非交换的结合代数，或微分渐变交换代数。一类重要的非交换流形可以通过交换代数的变形得到。变形量子化理论处于经典力学和量子力学的边界上。这两种理论的数学结构非常不同。量子化，粗略地说，是对量子现象的研究和预测，这些现象通常由非交换结合代数描述，来自它们潜在的经典对立物的几何。本项目将重点研究非交换几何中的同伦代数结构，使用变形量化、李群和李代数理论等工具。这些问题包括探索Duflo和Todd型类，建立kontsevic -Duflo型定理，以及在李代数的一般框架下研究Tsygan非交换微积分。

项目成果

Fedosov dg manifolds associated with Lie pairs

• DOI: 10.1007/s00208-020-02012-6
• 发表时间: 2016-05
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Mathieu Sti'enon;P. Xu

Atiyah classes and Kontsevich–Duflo type theorem for dg manifolds

• DOI: 10.4064/bc123-3
• 发表时间: 2021
• 期刊: Banach Center Publications
• 作者: M. Stiénon;P. Xu