Questions on Algebraic Operads and Related Structures

运算及其推广用于描述一大类代数结构。它们在研究各种几何对象、解决量子力学的基础问题以及解决数论中的某些问题方面发挥着重要作用。该研究项目旨在解决这一领域的重要开放问题。除了追求研究项目，调查员将建议研究生的研究项目，并期望监督本科生的研究项目以及。与合作者，调查员正在编写一本研究生水平的教科书变形量子化的辛流形。主要研究人员正在解决的Deligne-Drinfeld猜想上的Grothendieck-Teichmueller李代数使用变形复杂的运算支配Gerstenhaber代数和Kontsevich的图形复杂的变形量子化。PI正在研究一系列与模运算有关的问题，这些问题是通过将费曼变换应用于控制交换代数的运算而获得的。PI还研究了分次基上的变形量子化问题，并研究了同伦代数上的更高范畴结构。Grothendieck-Teichmueller李代数的研究是由它与变形量子化、有理数的绝对伽罗瓦群和动机理论的联系所激发的。关于控制交换代数的运算的费曼变换的问题是由长结点空间的研究引起的。

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.