Homotopy Algebras in Noncommutative Geometry

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This project concerns the investigation of problems in noncommutative geometry. The idea of noncommutative geometry is to study geometry using noncommutative algebras, which are mathematical objects that have operations of addition and multiplication; however, the multiplication may not be commutative: xy does not have to equal yx. The purpose of the project is to investigate a set of mathematical problems motivated by physics. More specifically, the motivation derives from a combination of ideas from quantum mechanics, string theory, and classical areas of mathematics such as algebra and geometry. The interdisciplinary nature of the research promotes further interaction between these fields. The project provides excellent opportunities for the investigator to work with young scientists and to exchange ideas with colleagues from other countries to promote scientific collaborations.In noncommutative geometry, one studies geometry via algebras of functions on noncommutative manifolds. On such a noncommutative manifold, the relevant objects are no longer points in a space, but rather a noncommutative associative algebra, or a differential graded commutative algebra. An important class of noncommutative manifolds can be obtained as deformations of commutative algebras. The theory of deformation quantization lies on the boundary between classical and quantum mechanics. The mathematical structures of the two theories are very different. Quantization, roughly speaking, is the study and prediction of quantum phenomena, which are normally described by noncommutative associative algebras, from the geometry of their underlying classical counterparts. This project will focus on the study of homotopy algebra structures in noncommutative geometry using tools from deformation quantization and Lie groupoid and Lie algebroid theory. The problems include exploring the Duflo and Todd type class, establishing a Kontsevich-Duflo type theorem, and studying Tsygan noncommutative calculi in a general framework in terms of Lie algebroids.