Algebraic Topology and Algebraic K-theory

代数拓扑和代数 K 理论

• 批准号: 1505579
• 负责人: Michael Mandell
• 金额: $ 20.5万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1505579&HistoricalAwards=false
• 关键词: Algebraic; Topology; theory

This project uses homotopy theory to study questions in number theory and in algebraic and differential topology. 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.The project has three main foci. The first focus concerns the algebraic K-theory of the sphere spectrum, which connects to both differential topology and number theory. The research pursues a program to identify the homotopy type of the fiber of the linearization map between the K-theory of the sphere spectrum and the K-theory of the integers; a program to identify the homotopy fiber of the cyclotomic trace; and an approach to a conjecture of Hesselholt by constructing an appropriate spectral category of cyclotomic spectra. The second focus is the study of the complex cobordism spectrum as an E_infinity or more generally E_n ring spectrum. The project investigates some concrete questions on the multiplication on cobordism and approaches to solving them. The third focus concerns the relationship between the E_n differential graded algebras and the (unstable) homotopy theory of simply connected spaces. The work includes a project for exploring the notion of "formal" E_n algebra, particularly the question of when the cochain complex of a space can be formal and the related question of when it is equivalent to a commutative differential graded algebra.

这个项目使用同伦理论来研究数论和代数与微分拓扑学中的问题。 同伦理论研究的是那些在小变形下不发生变化的数学对象的性质。 这些数学对象通常具有几何性质，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。 第一个焦点是球谱的代数K-理论，它与微分拓扑和数论都有联系。 研究了球谱的K-理论与整数的K-理论之间的线性化映射的纤维的同伦类型的识别程序;分圆迹的同伦纤维的识别程序;以及通过构造适当的分圆谱的谱范畴来解决Hesselholt猜想的方法。第二个重点是研究作为E_∞或更一般的E_n环谱的复配边谱。 本文研究了配边乘法的一些具体问题及其解决方法。 第三个焦点是E_n微分分次代数与单连通空间的（不稳定）同伦理论之间的关系。 其中包括一个探索“形式”E_n代数概念的项目，特别是空间的上链复形何时可以是形式的问题，以及与之相关的何时等价于交换微分分次代数的问题。