Questions on Algebraic Operads and Related Structures

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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.