Mathematical Sciences: Cyclic Homology and Differential Algebra

这个奖项的获奖者的初步研究旨在描述循环同调的代数品种使用的技术在微分代数。 在早期的工作中，主要的修正者已经表明，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，即米尔诺尔数或丘里纳数。 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表示。 主算子也将作用于具有孤立奇点的曲线上的微分算子的运动。 本研究涉及代数K-理论。 从广义上讲，代数K-理论涉及线性代数中基、向量空间等概念的演化。 这项工作有重大的影响，数论和代数几何，并承诺作出令人兴奋的连接之间的一些不同领域的数学。

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.