Questions on Algebraic Operads and Related Structures

操作数及其泛化用于描述一大类代数结构。它们在研究各种几何物体、解决量子力学的基本问题以及解决数论中的某些问题方面发挥着重要作用。本研究项目旨在解决这一领域的重要开放性问题。除了进行研究项目外，研究员还将为研究生的研究项目提供建议，并期望指导本科生的研究项目。与合著者共同撰写研究生水平的辛流形变形量化教科书。首席研究员正在利用控制Gerstenhaber代数的算子的变形复合体和与变形量化相关的Kontsevich图复合体来解决Grothendieck-Teichmueller李代数上的Deligne-Drinfeld猜想。PI正在研究与模操作数相关的一系列问题，模操作数是通过对控制交换代数的操作数应用费曼变换得到的。PI还研究了梯度基上的变形量子化问题，并研究了同伦代数上的高范畴结构。Grothendieck-Teichmueller李代数的研究是由其与变形量子化、绝对伽罗瓦有理数群和动机理论的联系所推动的。关于控制交换代数的算子的费曼变换的问题是由长结空间的研究引起的。

Operads and their generalizations are used to describe a large class of algebraic structures. They play an important role in the study of various geometric objects, in tackling foundational questions of quantum mechanics, and in addressing certain questions in number theory. This research project aims to resolve important open questions in this area. In addition to pursuing the research project, the investigator will advise graduate student research projects and expects to supervise undergraduate research projects as well. Jointly with coauthors, the investigator is writing a graduate level textbook on deformation quantization of symplectic manifolds. The principal investigator is tackling the Deligne-Drinfeld conjecture on the Grothendieck-Teichmueller Lie algebra using the deformation complex of the operad governing Gerstenhaber algebras and Kontsevich's graph complex related to deformation quantization. The PI is working on a circle of problems related to the modular operad which is obtained by applying the Feynman transform to the operad governing commutative algebras. The PI also works on the problem of deformation quantization over a graded base and studies a higher categorical structure on homotopy algebras. The study of the Grothendieck-Teichmueller Lie algebra is motivated by its links to deformation quantization, the absolute Galois group of rational numbers and the theory of motives. Questions about the Feynman transform of the operad governing commutative algebras are motivated by the study of spaces of long knots.