This project involves problems in noncommutative geometry. The idea of noncommutative geometry is to study geometry using noncommutative algebras, which are mathematical objects that have addition and multiplication, however the multiplication is not commutative: xy does not have to equal yx. The purpose of the project, which is centered around higher structures and noncommutative geometry, is to investigate mathematical problems in these fields motivated by physics. More specifically, it is motivated by a combination of ideas from quantum mechanics, quantum field theory, string theory, deformation quantization, differential geometry, noncommutative geometry, and operator algebras. 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. Some parts of the project are joint work with collaborators from China and Europe. The project thus provides an excellent opportunity for the PI to work with young scientists and to exchange ideas with colleagues from other countries and to promote collaborations.
The idea of noncommutative geometry is to study 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. Many noncommutative spaces can be obtained from groupoids. The PI proposes to continue the study of deformation quantization, differential graded manifolds (DG manifolds), Chern--Connes character, and delocalized twisted equivariant cohomology. The problems include exploring the homotopy Lie algebra structures appearing in connection with a generalization of the Atiyah class to various contexts, and developing a de Rham model of equivariant twisted K-theory using the theory of Lie groupoids and differentiable stacks.
