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

本文研究了交换代数变形的非交换微分几何；微分算子的代数；伪微分算子的符号；量子群与空间的坐标代数以及量子流形上的微分算子。首席研究员将通过他对量子群和空间的循环同调理论的研究，发展量子流形的黎曼-洛克定理。这些结果在算术和共形量子场论中的应用也将被探讨。本课题的研究是流形微积分的一个衍生。粗略地说，流形是曲面和它们的高维类似物。微分算子来自于微积分，它只能看到流形的小块，但却包含了流形整体形状和构象的信息。从微分算子中提取这类信息的一种策略是将算子组合成代数结构，然后使用代数机制。这个项目与这种机器的发展有关。

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.