Higher Structures, Homotopy Algebras, and Noncommutative Geometry

这个项目涉及到非对易几何的问题。非对易几何的思想是利用受几何启发的工具来研究非对易代数。非交换代数是具有加法和乘法的数学对象；然而，元素相乘的顺序可能很重要。该项目以非对易几何中的更高结构和同伦代数为中心，目的是研究这些领域中由物理驱动的数学问题。更具体地说，这个项目是由量子力学、量子场论、弦理论和经典数学领域(如李论、表示论、复几何、同调代数、分层理论、形变量子化和指数理论以及非对易几何)的组合推动的。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。国际和平协会继续通过在会议和研讨会上发言和组织研讨会来传播他的研究成果，这些研讨会为国际和平协会提供了与来自美国国内外的同事，特别是年轻科学家交流、互动和合作的绝佳机会。该奖项将支持在相关领域工作的早期职业研究人员的培训。在非交换几何中，人们研究代数就像它们是流形上的函数的代数一样。然而，这些非对易空间是虚拟的，不是由点组成的。DG流形是这种非对易空间的一种特殊类型。Dg流形上的函数空间是微分分次代数。作为交换代数的变形，得到了另一类重要的非交换空间。形变量子化旨在为经典力学和量子力学之间架起一座桥梁。这两个理论的数学结构非常不同，这使得理解从经典到量子的转换是如何工作的一个具有挑战性的问题。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代数描述，来自其潜在的经典对应的几何。PI建议利用形变量子化和李代数体理论的工具，继续研究非对易几何中自然产生的更高结构和同伦代数及其与表示论的关系。这些问题包括调查托德类在与dg流形相关的Tamarkin-Tsygan演算中的作用，研究dg流形的形式几何和dg李代数群的同伦等价概念，在广泛的背景下建立Kontsevich-Duflo类型定理，以及探索负分级dg流形。该奖项反映了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。

This project involves problems in noncommutative geometry. The idea of noncommutative geometry is to study noncommutative algebras using tools inspired by geometry. Noncommutative algebras are mathematical objects that have addition and multiplication; however the order in which the elements get multiplied might matter. The purpose of the project, which is centeredaround higher structures and homotopy algebras in noncommutative geometry, is to investigate mathematical problems motivated by physics in these fields. More specifically, the project is motivated by a combination of ideas from quantum mechanics, quantum field theory, string theory, and classical areas of mathematics such as Lie theory, representation theory, complex geometry, homological algebra, foliation theory, deformation quantization and index theory, and noncommutative geometry. 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, which provide excellent opportunities for the PI to exchange, interact and collaborate with colleagues from within and outside the US and, in particular, young scientists. This award will support the training of early career researchers that work on related fields.In noncommutative geometry one studies algebras as if they were algebras of functions on manifolds. However, these noncommutative spaces are virtual and not made of points. Dg manifolds are one particular type of such a noncommutative spaces. The space of functions on a dg manifold is a differential graded algebra. Another important class of noncommutative spaces is obtained as deformations of commutative algebras. Deformation quantization aims at throwing a bridge between classical and quantum mechanics. The mathematical structures of the two theories are very different, making it a challenging problem to understand how the transition from classical to quantum works. Quantization, roughly speaking, is the study and prediction of quantum phenomena, which are normally described by noncommutative algebras, from the geometry of their underlying classical counterparts. The PI proposes to continue the study of higher structures and homotopy algebras arising naturally in noncommutative geometry and their relation to representation theory using tools from deformation quantization and Lie algebroid theory. The problems include investigating the role of the Todd class in Tamarkin-Tsygan calculi associated with a dg manifold, studying the formal geometry of dg manifolds and the concept of homotopy equivalence of dg Lie algebroids, establishing a Kontsevich-Duflo type theorem in a wide context, and exploring negatively graded dg manifolds.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.