This grant proposes to use E-infinity differential graded algebras to study the homotopy theory of spaces and proposes to study the homotopy theory of E-infinity differential graded algebras and E-infinity ring spectra. The concept of E-infinity differential graded algebra is a generalization of the concept of commutative differential graded algebra; commutativity holds only up to homotopy, and E-infinity differential graded algebras admit Steenrod operations on their cohomology, which measure to some extent the deviation from actual commutativity. E-infinity ring spectra are a stable homotopy theory generalization of E-infinity differential graded algebras. Most commonly studied generalized cohomology theories that have commutative ring structures are represented by E-infinity ring spectra. This extra structure yields important calculational information about the theory and is useful for various constructions of related theories.
Algebraic topology tries to reduce topological or geometric classification problems into algebra. Because the algebra is usually discrete, it is typically invariant under changes by continuous deformations, or ``homotopies.'' Because of this, fundamental problems in algebraic topology are often phrased in terms of understanding the homotopy equivalence classes of spaces or in terms of computing the number (or more often the ``group'' or ``module'') of homotopy classes of maps between spaces. Quite generally this kind of question can be reformulated as an equivalent question in algebra. The purpose of this proposal is to develop tools to study the sort of algebra that arises in this context.