Mathematical Sciences: Cyclic Homology and Differential Algebra

该奖项支持旨在利用微分代数技术描述代数变种的循环同调的初步研究。在早期的工作中，主要研究者已经证明了循环同调可以用来恢复具有孤立奇点的平面曲线的经典不变量，即密尔诺数或朱里纳数。她现在将致力于获得一般代数曲线的hochschild同调和循环同调的显式计算。首席研究员还将研究具有孤立奇点曲线上的微分算子模块。本研究涉及代数k理论。从广义上讲，代数k理论涉及线性代数概念的演变，如基空间和向量空间。这项工作对数论和代数几何具有重要意义，并有望在许多不同的数学领域之间建立令人兴奋的联系。

THIS AWARD SUPPORTS PRELIMINARY RESEARCH AIMED AT A DESCRIPTION OF CYCLIC HOMOLOGY OF ALGEBRAIC VARIETIES USING TECHNIQUES IN DIFFERENTIAL ALGEBRA. IN EARLIER WORK THE PRINCIPAL INVESTIGATOR HAS SHOWN THAT CYCLIC HOMOLOGY CAN BE USED TO RECOVER A CLASSICAL INVARIANT, THE MILNOR OR TJURINA NUMBER, OF PLANE CURVES WITH ISOLATED SINGULARITIES. SHE WILL NOW WORK ON OBTAINING AN EXPLICIT COMPUTATION OF HOCHSCHILD HOMOLOGY AND CYCLIC HOMOLOGY FOR GENERAL ALGEBRAIC CURVES. THE PRINCIPAL INVESTIGATOR WILL ALSO WORK ON THE MODULE OF DIFFERENTIAL OPERATORS ON CURVES WITH ISOLATED SINGULARITIES. THIS RESEARCH IS CONCERNED WITH ALGEBRAIC K-THEORY. IN A BROAD SENSE, ALGEBRAIC K-THEORY CONCERNS THE EVOLUTION OF CONCEPTS FROM LINEAR ALGEBRA SUCH AS BASIS AND VECTOR SPACE. THIS WORK HAS SIGNIFICANT IMPLICATIONS FOR NUMBER THEORY AND ALGEBRAIC GEOMETRY, AND PROMISES TO MAKE EXCITING CONNECTIONS BETWEEN A NUMBER OF DIFFERENT AREAS OF MATHEMATICS.