Differential geometry, noncommutative geometry and quantization

微分几何、非交换几何和量子化

• 批准号: 0604552
• 负责人: Xiang Tang
• 金额: $ 6.54万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2006-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604552&HistoricalAwards=false
• 关键词: Differential; geometry; noncommutative; quantization

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.

唐工程的几个问题，微分几何和非交换几何。主要是他应用的方法和思想，从非交换几何的研究微分几何，反之亦然。Tang从真Etale广群及其广群代数的角度研究轨道折叠。他计算Hochschild和循环上同调的变形量化这些广群代数。他正在进一步研究Gerstenhaber代数结构上的Hochschild上同调的广群代数，以解决金兹伯格-卡列丁猜想上的陈阮orbifold上同调。作为对orbifolds研究的延伸，Tang将研究更复杂的商奇点。特别是，他将继续他的研究单位的连接groupoid和堆栈。非交换几何在微分几何中的另一个应用是代数指标定理。变形量子化的代数指标定理由Fedosov-Nest-Tsygan发展。唐将运用他们的思想来研究指标问题的orbifolds和量化的接触变换。在另一个方向上，应用从微分几何到非交换几何的技术，唐研究了康纳斯和莫斯科维奇的霍普夫代数的兰金-科恩变形，它最初是在数论中的模形式上构造的。主要的几何输入是一个叶理的叶子空间的辛几何。 辛几何和数论之间的联系也将被调查。唐正致力于发展一个概念的hopfish代数作为一个推广的霍普夫代数。众所周知，非交换环面代数不是Hopf代数，然而，在非交换环面代数上已经发现了Hopfish结构的候选者。将继续对该结构进行分析。最后，将研究非交换超几何和复几何，例如Q-代数和规范理论，非交换复流形的例子。唐的研究涉及两个数学领域之间的相互作用，微分几何和非交换几何。微分几何提供了经典物理学的数学公式，非对易几何为量子物理学提供了严格的基础。类似于经典物理和量子物理之间的关系，微分几何的发展为非对易几何的研究提供了工具、直觉和灵感，反之亦然。一方面，Tang使用微分几何来理解量子群的变形，量子群最初是从数论中的模形式构造的，并发展了更一般的量子对称性概念，并寻找更多的非对易超流形和复流形的例子;在另一个方向，使用非对易几何学中开发的工具，唐研究微分几何的问题，这是很难解决的经典几何工具，如orbifolds和奇异空间，和各种指标的问题。