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

这个奖项是关于交换环变形的非交换微分几何的研究。主要研究者证明了辛流形上函数环的任意变形的代数指标定理。主要的技术工具是循环同调中的运算理论。他计划进一步发展这一构造，这将产生族和叶的代数指标定理，函数环的等变量化的cones - moscovici指标定理的代数模拟，以及交换环量化的指标和特征公式表示。这些方法也有望产生黎曼-罗赫-格罗滕迪克定理的各种推广。这是代数几何领域的研究，是现代数学中最古老的部分之一，但在过去的25年里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，而且还使用分析和拓扑的方法，相反，它在这些领域以及理论计算机科学和机器人技术中也得到了应用。

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.