Applications of Higher Algebraic Structures in Noncommutative Geometry

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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.