Homotopy Algebras and Homotopy Theory

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The goal of this project is to explore some applications and potential applications of homotopy algebras. In algebraic K-theory, the proposal describes ideas applicable to the conjectures of Rognes and the conjecture of Waldhausen on the K-theory chromatic tower, relating arithmetic and geometry. In unstable homotopy theory, the proposal describes how E-infinity differential graded algebras and related homotopy algebras can be used to study the homotopy theory of spaces. In stable homotopy theory, the proposal discusses obstruction theory for E-n algebra structures, and the relationship of E-n algebra structures with structures on categories of modules.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.