Mathematical Sciences: Non-commutative Differential Geometryof Deformations of Commutative Rings: Operations Index Theorems and Characteristic Classes

这个奖项是关于非交换的研究 交换环变形的微分几何。 的 首席研究员证明了代数指数定理， 辛上函数环的任何变形 歧管 主要的技术工具是作战理论 在循环同源中。 他计划开发这座建筑 此外，这将产生代数指数定理， 家庭和叶理，代数模拟的 等变量子化的Connes-Moscovici指标定理 一个函数环，以及指数和特征标公式 交换环的量子化的表示。 这些 方法也有望产生各种概括的 Riemann-Roch-Grothendieck定理 这在代数几何领域是合理的， 现代数学中最古老的部分，但其中有一个 在过去的四分之一个世纪里，革命性的开花。 在其 起源，它处理的数字，可以定义在平面上， 最简单的方程，即多项式。 如今，该领域 不仅使用代数方法，而且使用分析方法， 拓扑学，反过来说， 以及理论计算机科学和机器人技术。

This award is concerned with the study of the noncommutative differential geometry of deformations of commutative rings. The principal investigator has proved the algebraic index theorem for any deformation of the ring of functions on a symplectic manifold. The main technical tool was the theory of operations in cyclic homology. He plans to develop this construction further, which would yield the algebraic index theorem for families and foliations, the algebraic analogue of the Connes-Moscovici index theorem for an equivariant quantization of a ring of functions, and index and character formula representations of quantizations of commutative rings. These methods are also expected to yield various generalizations of the Riemann-Roch-Grothendieck theorem. This is reasearch in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.