Higher Structures, Homotopy Algebras, and Noncommutative Geometry

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.