Homotopy Algebras and Homotopy Theory

同伦代数和同伦理论

• 批准号: 0504069
• 负责人: Michael Mandell
• 金额: $ 10.92万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-05-01 至 2009-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504069&HistoricalAwards=false
• 关键词: Homotopy; Algebras; Theory

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.

该项目的目标是探索同伦代数的一些应用和潜在应用。 在代数拓扑中，E-无穷微分分级代数和相关的同伦代数可以用来研究空间的同伦理论。 这种同类结构的其他版本可以潜在地应用于代数几何中，以在同伦不变量上放置“动机结构”。 E_2代数和幺半群范畴之间有密切的关系。 E_3代数和辫状幺半群范畴之间应该有类似的密切关系。 该项目还研究置换范畴的代数K理论以及置换范畴与稳定同伦理论之间的密切联系。同伦理论研究数学对象在小变形下不改变的那些属性。 这些数学对象通常具有几何性质，但同伦论的方法也越来越多地应用于代数对象。 通过利用小变化下的不变性，同伦论性质往往可以进行计算。 由于它们通常还保留有关原始数学对象的重要信息，因此同伦理论是解决各种数学问题的有效工具。