Differential geometry, noncommutative geometry and quantization

唐研究微分几何和非交换几何中的几个问题。他主要将非交换几何的方法和思想应用于微分几何的研究，反之亦然。唐从适当的群拟及其群拟代数的角度研究轨道。他计算了这些类群代数的变形量子化的Hochschild和循环上同。他正在进一步研究群拟代数的Hochschild上同调上的Gerstenhaber代数结构，以解决Chen-Ruan轨道上同调上的Ginzburg-Kaledin猜想。作为对轨道研究的延伸，Tang将研究更复杂的商奇点。特别是，他将继续研究群拟和堆栈上的平连接。非交换几何在微分几何中的另一个应用涉及代数指数定理。Fedosov-Nest-Tsygan提出了变形量化的代数指标定理。唐将把他们的想法应用于研究轨道和量化接触变换上的指数问题。在另一个方向上，运用从微分几何到非交换几何的技术，Tang研究了最初在数论中基于模形式构造的Hopf代数的Connes和Moscovici的Rankin-Cohen变形。主要的几何输入是叶理的叶空间的辛几何。辛几何和数论之间的联系也将被研究。Tang正致力于发展hopfish代数的概念，作为Hopf代数的推广。已知非交换环面代数不是Hopf代数，但在非交换环面代数上发现了hopfish结构的候选结构。对这种结构的分析还将继续。最后，研究非交换的超几何和复几何，例如q代数和规范理论，非交换的复流形的例子。唐的研究涉及数学的两个领域，微分几何和非交换几何之间的相互作用。微分几何提供了经典物理的数学公式，而非交换几何为量子物理提供了严格的基础。与经典物理和量子物理之间的关系类似，微分几何的发展为非交换几何的研究提供了工具、直觉和灵感，反之亦然。在一个方向上，Tang使用微分几何来理解量子群的变形，量子群最初是由数论中的模形式构造的，并发展更一般的量子对称概念，并寻找更多的非交换超流形和复流形的例子；另一方面，利用非交换几何中发展起来的工具，研究了用经典几何工具难以解决的微分几何问题，如轨道和奇异空间，以及各种指标问题。

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.