Homotopy Algebras and Homotopy Theory

这个项目的目的是探索同伦代数的一些应用和潜在的应用。在代数拓扑学中，E-无穷次微分分次代数及相关的同伦代数可用于研究空间的同伦论。这种结构的其他版本可以潜在地应用于基于同伦不变量的“动机结构”的代数几何中。E_2代数与么半群范畴之间有着密切的关系。E_3代数与辫子么半群范畴之间应该有类似的密切关系。该项目还研究了置换范畴的代数K-理论以及置换范畴与稳定同伦理论之间的密切联系。同伦理论研究了数学对象在小变形下不变的性质。这些数学对象通常是几何性质的，但同伦理论的方法也越来越多地应用于代数对象。同伦性质往往是通过利用微小变化下的不变性来进行计算的。由于同伦理论一般也保留了关于原始数学对象的重要信息，所以同伦理论是解决广泛数学问题的有效工具。

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.