Differential geometry, noncommutative geometry and quantization

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Tang works on several problems in differential geometry and noncommutative geometry. Mainly he applies the methods and ideas from noncommutative geometry to the study of differential geometry, and vice versa. Tang studies orbifolds from the point of view of proper etale groupoids and their groupoid algebras. He computed Hochschild and cyclic cohomology of the deformation quantization of these groupoid algebras. He is further studying the Gerstenhaber algebra structure on the Hochschild cohomology of the groupoid algebras to address the Ginzburg-Kaledin conjecture on the Chen-Ruan orbifold cohomology. As an extension to the study of orbifolds, Tang will investigate more complicated quotient singularities. In particular, he will continue his study of flat connections on groupoids and stacks. Another application of noncommutative geometry to differential geometry concerns algebraic index theorem. The algebraic index theorem of deformation quantization was developed by Fedosov-Nest-Tsygan. Tang will apply their ideas to study index problems on orbifolds and quantized contact transformations. In the other direction, applying techniques from differential geometry to noncommutative geometry, Tang studies Connes and Moscovici's Rankin-Cohen deformation of a Hopf algebra, which was originally constructed on modular form in number theory. The main geometric input is symplectic geometry of the space of leaves of a foliation. The connection between symplectic geometry and number theory will also be investigated. Tang is working on developing a notion of a hopfish algebra as a generalization of a Hopf algebra. It is known that a noncommutative torus algebra is not a Hopf algebra, however, a candidate for a hopfish structure on a noncommutative torus algebra has been discovered. The analysis of this structure will be will continued. Finally, noncommutative super geometry and complex geometry will be investigated, e.g. Q-algebras and gauge theory, examples of noncommutative complex manifolds. Tang's research concerns the interplay between two fields of mathematics, differential geometry and noncommutative geometry. Differential geometry provides a mathematical formulation of classical physics, and noncommutative geometry gives a rigorous foundation for quantum physics. Analogous to the relation between classical and quantum physics, the development of differential geometry provides tools, intuition, and inspiration for the study of noncommutative geometry and vice versa. In one direction, Tang uses differential geometry to understand deformation of quantum groups, which was originally constructed from modular forms in number theory, and to develop more general notion of quantum symmetry, and to look for more examples of noncommutative supermanifolds and complex manifolds; in the other direction, using tools developed in noncommutative geometry, Tang studies problems in differential geometry which are hard to solve using classical geometry tools, e.g. orbifolds and singular spaces, and various index problems.