Applications of Higher Algebraic Structures in Noncommutative Geometry

本项目涉及非交换几何中的问题；也就是说，使用受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。该项目的目的是围绕非交换几何中的高等代数结构和同伦代数，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。PI将继续研究高等代数结构和同伦代数在非交换几何中的应用，以及它们与表示理论的关系，使用变形量化和李代数理论的工具。本文将建立一个同伦李代数群的kontsevic - duflo型定理，建立一个同伦kontsevic - soibelman结构的形式态射，以及dg李代数群的Tamarkin-Tsygan非交换演算。更具体地说，该项目包含了几个独立的相关问题，包括构造同伦李代数的普遍包络代数，研究同伦kontsevic - soibelman结构和多微分算子和多射流空间对的Tamarkin-Tsygan微积分，以及与dg李代数相关的多向量场和微分形式的空间对，以及发展dg李代数的形式几何。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 concerns problems in noncommutative geometry; that is, the study of 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 centered around higher algebraic 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.The PI will continue the study of applications of higher algebraic structures and homotopy algebras in noncommutative geometry and their relation to representation theory using tools from deformation quantization and Lie algebroid theory. The PI will establish a Kontsevich-Duflo type theorem for homotopy Lie algebroids and will establish a formality morphism for homotopy Kontsevich-Soibelman structures and the noncommutative calculi à la Tamarkin-Tsygan for dg Lie algebroids. More specifically, the project encompasses several related problems of independent interest including constructing the universal enveloping algebra of a homotopy Lie algebroid, studying homotopy Kontsevich-Soibelman structures and the Tamarkin-Tsygan calculi for the pair of spaces of polydifferential operators and polyjets as well as the pair of spaces of polyvector fields and differential forms associated to a dg Lie algebroid, and developing a formal geometry for dg Lie algebroids.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.