Algebraic Topology and Algebraic K-theory

Indiana University, Bloomington IN

Investigators

Abstract

This project uses homotopy theory to study questions in number theory and in algebraic and differential topology. Homotopy theory studies those properties of mathematical objects that do not change under small deformations. These mathematical objects are often of a geometric nature, but the methods of homotopy theory have been increasingly applied to objects of an algebraic nature as well. Homotopy theoretic properties tend to be accessible to computation by taking advantage of the invariance under small changes. Since they also generally retain important information about the original mathematical objects, homotopy theory is an effective tool for a wide range of mathematical problems. The project has three main foci. The first focus concerns the algebraic K-theory of the sphere spectrum, which connects to both differential topology and number theory. The research pursues a program to identify the homotopy type of the fiber of the linearization map between the K-theory of the sphere spectrum and the K-theory of the integers; a program to identify the homotopy fiber of the cyclotomic trace; and an approach to a conjecture of Hesselholt by constructing an appropriate spectral category of cyclotomic spectra. The second focus is the study of the complex cobordism spectrum as an E_infinity or more generally E_n ring spectrum. The project investigates some concrete questions on the multiplication on cobordism and approaches to solving them. The third focus concerns the relationship between the E_n differential graded algebras and the (unstable) homotopy theory of simply connected spaces. The work includes a project for exploring the notion of "formal" E_n algebra, particularly the question of when the cochain complex of a space can be formal and the related question of when it is equivalent to a commutative differential graded algebra.