Homotopy Algebras and Homotopy Theory

点击翻译按钮获取中文摘要

The goal of this project is to explore some applications and potentialapplications of homotopy algebras. In algebraic topology, E-infinitydifferential graded algebras and related homotopy algebras can be usedto study the homotopy theory of spaces. Other versions of this samekind of structure can potentially be applied in algebraic geometry toput ``motivic structures'' on homotopy invariants. There is a closerelationship between E_2 algebras and monoidal categories. Thereshould be an analogously close relationship between E_3 algebras andbraided monoidal categories. The project also studies the algebraicK-theory of permutative categories and the close ties betweenpermutative categories and stable homotopy theory.Homotopy theory studies those properties of mathematical objects thatdo not change under small deformations. These mathematical objectsare often of a geometric nature but the methods of homotopy theoryhave been increasingly applied to algebraic ones as well. Homotopytheoretic properties tend to be accessible to computation by takingadvantage of the invariance under small changes. Since they alsogenerally retain important information about the original mathematicalobjects, homotopy theory is an effective tool for a wide rangeof mathematical problems.