Associate professor of mathematics
Professor McCarthy has been a member of the Department of Mathematics since 1994. His research has been primarily in algebraic topology and algebraic K-theory. During his Center appointment, he will investigate a new filtration of algebraic K-theory obtained by extending ideas used in the calculus of homotopy functors. Classically, calculus is a systematic study of functions that attempts to use local information like the derivative to predict global behavior. One of the great successes of calculus is the Taylor series. Given a function, the Taylor series gives an algorithmic formula for a sequence of polynomials that approximates the function, assuming one knows the derivatives of the function at some fixed point. In topology, one often studies a generalization of a function, called a homotopy functor, that takes topological spaces to topological spaces instead of real numbers to real numbers. In the last decade, a generalization of the Taylor series for functions has been used in the study of homotopy functors, and the resulting calculus theory has been successfully applied to the study of algebraic K-theory. Recently, McCarthy has generalized the calculus of homotopy functors and also developed a dual calculus. This dual calculus measures a piece of algebraic K-theory important to topological obstruction theory that cannot be effectively detected by the usual calculus of homotopy functors. The main goal for this project is to construct a computable algebraic model for the dual derivative of algebraic K-theory similar to the way in which topological Hochschild homology models the derivative of algebraic K-theory. By standard techniques, this will give information for all the layers of the dual tower, which will provide a new tool for making explicit calculations where currently very few exist.