Homotopy Algebras and Homotopy Theory

Indiana University, Bloomington IN

Investigators

Abstract

The goal of this project is to explore some applications and potential applications of homotopy algebras. In algebraic topology, E-infinity differential graded algebras and related homotopy algebras can be used to study the homotopy theory of spaces. Other versions of this same kind of structure can potentially be applied in algebraic geometry to put ``motivic structures'' on homotopy invariants. There is a close relationship between E_2 algebras and monoidal categories. There should be an analogously close relationship between E_3 algebras and braided monoidal categories. The project also studies the algebraic K-theory of permutative categories and the close ties between permutative categories and stable homotopy theory. 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 algebraic ones 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.