Higher Structures, Homotopy Algebras, and Noncommutative Geometry

这个项目涉及非交换几何中的问题。非交换几何的思想是利用几何启发的工具来研究非交换代数。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。该项目以非交换几何中的高等结构和同伦代数为中心，目的是研究这些领域中由物理引起的数学问题。更具体地说，该项目是由量子力学、量子场论、弦理论和经典数学领域（如李论、表示理论、复几何、同调代数、叶理理论、变形量子化和指标理论以及非交换几何）的思想结合而成的。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研讨会来传播他的研究，这为PI提供了与美国国内外同事，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，人们把代数当作流形上的函数代数来研究。然而，这些非交换空间是虚的，不是由点构成的。Dg流形是这种非交换空间的一种特殊类型。dg流形上的函数空间是一个微分渐变代数。另一类重要的非交换空间是可交换代数的变形。变形量子化的目的是在经典力学和量子力学之间架起一座桥梁。这两种理论的数学结构非常不同，这使得理解从经典到量子的转变是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，这些现象通常由非交换代数描述，来自它们潜在的经典对立物的几何。PI建议使用变形量化和李代数理论的工具继续研究非交换几何中自然产生的高级结构和同伦代数及其与表示理论的关系。这些问题包括研究Todd类在与dg流形相关的Tamarkin-Tsygan微积分中的作用，研究dg流形的形式几何和dg Lie代数的同伦等价概念，在广泛的背景下建立kontsevic - duflo型定理，以及探索负梯度dg流形。该奖项反映了美国国家科学基金会的法定使命，并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，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.