Mathematical Sciences: Homotopy Theory of Spectra

数学科学：谱的同伦论

• 批准号: 9505130
• 负责人: Jeffrey Smith
• 金额: $ 5万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9505130&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Homotopy; Theory; Spectra

9505130 Smith Spectra have been important in homotopy theory for many years. Recently the investigator has constructed a category of spectra with a symmetric monoidal smash product. The construction is elementary and will provide an introduction to spectra which will not scare away the novice. The strict associativity of the smash product means that many other definitions regarding spectra can be greatly simplified. For example, one can easily discuss ring spectra, commutative ring spectra, module spectra. The investigator is using this new definition of ring spectra in the study of topological Hochschild homology and topological cyclic homology. Spectra have been developed over the last thirty years as a bridge between the geometric problems that the algebraic topologist studies and the algebraic techniques which he uses. The earliest examples were given by Rene Thom in his work on the cobordism classification of manifolds, where he showed that manifolds, the geometric objects common to mathematics and physics, are classified up to cobordism by certain spectra. Over the years, as one needed spectra having more and more properties, many different constructions have been given. These constructions have became increasingly complicated, until only a few experts could understand them. However, recently the investigator has found a much easier way to construct spectra that nonetheless have all the properties one needs for the applications. This makes the field far more accessible, especially to graduate students, and should lead to much fuller appreciation of this powerful tool. ***

小行星9505130 谱在同伦理论中一直很重要。 最近，研究者构造了一类具有对称monoidal smash积的谱。 该结构是基本的，将提供一个介绍光谱，不会吓跑新手。 粉碎积的严格结合性意味着许多其他关于谱的定义可以大大简化。 例如，人们可以很容易地讨论环谱、交换环谱、模谱。 研究人员正在使用这个新的定义环光谱中， 拓扑Hochschild同调和拓扑循环同调研究 谱是在过去三十年中发展起来的，作为代数拓扑学家研究的几何问题和他使用的代数技术之间的桥梁。 最早的例子是雷内·托姆在他关于流形的配边分类的著作中给出的，在那里他展示了流形，数学和物理中常见的几何对象，通过某些谱被分类到配边。 多年来，随着人们对光谱性质的要求越来越高，人们给出了许多不同的光谱结构。 这些结构变得越来越复杂，直到只有少数专家才能理解它们。 然而，最近研究人员发现了一种更容易的方法来构建光谱，但具有应用所需的所有属性。 这使得该领域更容易获得，特别是对研究生来说，并应导致对这个强大工具的更全面的理解。 ***