Mathematical Sciences: Non-Commutative Differential Geometryof the Deformations of Commutative Rings

本文主要研究交换代数的变形、微分算子的代数、伪微分算子的符号、量子群和空间的坐标代数以及量子流形上的微分算子的非对易微分几何。主要研究人员将通过他对量子群和空间的循环同调理论的工作来发展量子流形的Riemann-Roch定理。这些结果在算术和共形量子场论中的应用也将被探索。这个项目的研究是微积分在流形上的产物。粗略地说，流形是表面和它们的高维类似物。微分运算符来自微积分，它只能看到流形的小片段，但仍然包含关于其整体形状和构象的信息。从微分运算符中提取这类信息的一种策略涉及将运算符组装成代数结构，然后使用代数机器。这个项目与这台机器的发展有关。

This research is concerned with the noncommutative differential geometry of the deformations of commutative algebras; algebras of differential operators; symbols of pseudo-differential operators; coordinate algebras of quantum groups and spaces; and differential operators on quantum manifolds. The principal investigator will develop a Riemann-Roch theorem for quantum manifolds through his work on the cyclic homology theory of quantum groups and spaces. Applications of these results to arithmetic and conformal quantum field theory will also be explored. The research in this project is an outgrowth of calculus on manifolds. Roughly speaking, manifolds are surfaces and their higher-dimensional analogues. Differential operators, which come from calculus, see the manifold only in small pieces, but nevertheless contain information about its overall shape and conformation. One strategy for extracting this sort of information from differential operators involves assembling the operators into algebraic structures and then using algebraic machinery. This project is concerned with the development of this machinery.