E-infinity Algebras and Homotopy Theory

DMS-0203980 Michael Mandell A.本基金拟利用E-无穷微分分次代数研究空间的同伦理论，拟研究E-无穷微分分次代数的同伦理论和E-无穷环谱。 E-无穷微分分次代数的概念是交换微分分次代数概念的推广;交换性只适用于同伦，并且E-无穷微分分次代数允许在其上同调上进行Steenrod运算，这在一定程度上度量了与实际交换性的偏差。 E-无穷环谱是E-无穷微分分次代数的稳定同伦理论的推广。 最常研究的具有交换环结构的广义上同调理论是用E-无穷环谱表示的。代数拓扑学试图把拓扑学或几何学的分类问题归结为代数问题，它是一种新的理论，它的结构可以产生关于理论的重要计算信息，并且对于相关理论的各种构造都很有用。 因为代数通常是离散的，所以它在连续变形或同伦的变化下通常是不变的。正因为如此，代数拓扑学中的基本问题经常被表述为理解空间的同伦等价类，或者计算空间之间映射的同伦类的数量（或者更常见的是“群”或“模”）。 一般来说，这类问题可以被重新表述为代数中的等价问题，本建议的目的是开发工具来研究在这种情况下出现的代数。

DMS-0203980Michael Mandell A.This grant proposes to use E-infinity differential graded algebras to study the homotopy theory of spaces and proposes to study the homotopy theory of E-infinity differential graded algebras and E-infinity ring spectra. The concept of E-infinity differential graded algebra is a generalization of the concept of commutative differential graded algebra; commutativity holds only up to homotopy, and E-infinity differential graded algebras admit Steenrod operations on their cohomology, which measure to some extent the deviation from actual commutativity. E-infinity ring spectra are a stable homotopy theory generalization of E-infinity differential graded algebras. Most commonly studied generalized cohomology theories that have commutative ring structures are represented by E-infinity ring spectra. This extrastructure yields important calculational information about the theory and is useful for various constructions of related theories.Algebraic topology tries to reduce topological or geometricclassification problems into algebra. Because the algebra is usually discrete, it is typically invariant under changes by continuous deformations, or ``homotopies.'' Because of this, fundamental problems in algebraic topology are often phrased in terms of understanding the homotopy equivalence classes of spaces or in terms of computing the number (or more often the ``group'' or ``module'') of homotopy classes of maps between spaces. Quite generally this kind of question can be reformulated as an equivalent question in algebra.The purpose of this proposal is to develop tools to study the sort of algebra that arises in this context.