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. ***

9505130 史密斯谱多年来一直在同伦理论中发挥着重要作用。 最近，研究人员构建了一类具有对称幺半群粉碎产物的光谱。 该结构很简单，并且将提供光谱介绍，不会吓跑新手。 粉碎乘积的严格关联性意味着可以大大简化有关光谱的许多其他定义。 例如，我们可以很容易地讨论环谱、交换环谱、模谱。 研究人员正在拓扑 Hochschild 同调和拓扑循环同调的研究中使用环谱的新定义。 谱在过去三十年中得到发展，成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 Rene Thom 在他关于流形的配边分类的著作中给出的，他在其中表明，流形（数学和物理学中常见的几何对象）可以通过某些谱分类为配边。 多年来，随着人们需要具有越来越多特性的光谱，已经给出了许多不同的结构。 这些结构变得越来越复杂，直到只有少数专家能够理解它们。 然而，最近研究人员发现了一种更简单的方法来构建光谱，但该方法具有应用所需的所有属性。 这使得这个领域更容易接触，特别是对于研究生来说，并且应该会导致人们更全面地认识这个强大的工具。 ***