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Formal group laws in homotopy theory and K-theory

同伦理论和 K 理论中的形式群定律

基本信息

批准号：
0805833
负责人：
Tyler Lawson
金额：
$ 12.8万
依托单位：
University of Minnesota-Twin Cities
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：
2008-06-01 至 2012-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805833&HistoricalAwards=false
关键词：
Formal group laws homotopy theory

项目摘要

AbstractAward: DMS-0805833Principal Investigator: Tyler D. LawsonThe goal of this project is to relate the theory of formal grouplaws to phenomena in stable homotopy theory and K-theory. Instable homotopy theory, the objective is to begin makingfundamental computations in the theory of topological automorphicforms developed in joint work with Behrens. This will firstproceed by examining and computing the homotopy of spectraassociated to certain moduli of abelian surfaces such as Shimuracurves, and then by working towards higher chromatic filtrations.In algebraic K-theory, the objective is to make systematic use ofthe relationship between formal group laws and complex cobordismto systematize computations in algebraic K-theory and relate themto the chromatic filtration, with the hope of gainingunderstanding of the "chromatic redshift" phenomenon.The subject of homotopy theory arose as a method to answerconcrete questions in geometry by way of algebra. In theprocess, a strange connection was discovered with formal grouplaws, which themselves are intimately related to quite differentsubjects such as number theory and elliptic curves. Inparticular, there is a surprising connection between ellipticcurves and mathematical physics. This research focuses onapplying very recent developments in homotopy theory to studyconnections such as these, with the hope that our understandingof number theory and our understanding of homotopy theory canbenefit each other.

摘要奖：DMS-0805833主要研究员：泰勒·D·劳森这个项目的目标是将形式群论与稳定同伦理论和K-理论中的现象联系起来。不稳定同伦理论，目标是开始在与Behrens共同发展的拓扑自同构形理论中进行基本计算。在代数K-理论中，目的是系统地利用形式群律和复余边律之间的关系来系统化代数K-理论中的计算，并将它们与色过滤联系起来，以期获得对“色红移”现象的理解。同伦理论是作为一种通过代数来回答几何中的具体问题的方法而产生的。在这个过程中，发现了一种与形式群规的奇怪联系，形式群规本身与完全不同的主题密切相关，如数论和椭圆曲线。特别是，椭圆曲线和数学物理之间有着惊人的联系。本研究的重点是将同伦论的最新发展应用于研究这样的联系，希望我们对数论的理解和对同伦论的理解能够互惠互利。