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.
