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

本研究关注的是非对易的 可换微分几何 代数;微分算子代数;符号 伪微分算子;坐标代数 量子群和空间;和微分算子 量子流形 首席研究员将制定一份 量子流形的Riemann-Roch定理通过他的工作 量子群和空间的循环同调理论。 这些结果在算术和保角上的应用 量子场论也将探讨。 这个项目的研究是微积分学的一个分支 在流形上。 粗略地说，流形是曲面， 它们的高维类似物。 微分算子， 它来自微积分，只在小的时候看到流形， 件，但仍然包含有关其整体 形状和构象。提取这种排序的一种策略是 来自微分算子的信息的集合 将运算符转化为代数结构，然后使用 代数机器 该项目涉及 这一机制的发展。

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.