Homotopy Algebras and Homotopy Theory

同伦代数和同伦理论

• 批准号: 0804272
• 负责人: Michael Mandell
• 金额: $ 11.65万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0804272&HistoricalAwards=false
• 关键词: Homotopy; Algebras; Theory

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.

这个项目的目标是探索同伦代数的一些应用和潜在的应用。在代数K理论中，该提案描述了适用于罗涅尼斯的猜想和瓦尔德豪森关于K理论色塔的猜想的思想，涉及算术和几何。在不稳定同伦理论中，该提案描述了如何使用E-无穷微分分次代数和相关的同伦代数来研究空间的同伦理论。在稳定同伦理论中，该提案讨论了E-n代数结构的阻塞理论，以及E-n代数结构与模范畴上的结构的关系。同伦理论研究数学对象在小变形下不发生变化的那些性质。 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性质的对象。 同伦理论的性质往往是可访问的计算，利用在小的变化下的不变性。由于同伦理论通常还保留了关于原始数学对象的重要信息，因此它是解决广泛数学问题的有效工具。