Formal group laws in homotopy theory and K-theory

摘要：项目负责人：Tyler D. lawson本项目的目的是将形式群法理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共同研究的拓扑自同构理论中进行基本计算。这将首先通过检查和计算与阿贝尔曲面（如shimuracurcurves）的某些模相关的光谱的同伦，然后通过向更高的色度滤波工作。在代数k理论中，目标是系统地利用形式群律和复协律之间的关系，将代数k理论中的计算系统化，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用代数来回答几何中具体问题的方法而产生的。在这个过程中，我们发现了一种与形式化群律之间的奇怪联系，形式化群律本身与数论和椭圆曲线等完全不同的学科密切相关。特别是，椭圆曲线和数学物理之间有着惊人的联系。本研究的重点是应用同伦理论的最新发展来研究诸如此类的联系，希望我们对数论的理解和对同伦理论的理解能够相互受益。

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.