Computational Chromatic Homotopy Theory

摘要奖：DMS 1612020，首席研究员：Agnes Beaudr这项资助的项目是理解球面同伦群的大规模努力的一部分，球面同伦群是代数拓扑学中的基本问题之一。球体是最简单的几何对象之一，是更复杂实体的构建块。球面的同伦群用于研究这些基本成分之间的联系，是被认为直到变形的球面之间的连续函数的集合。其他领域的许多问题，特别是微分拓扑学，已经归结为对这些群的研究，这是出了名的困难。然而，在代数几何和代数拓扑之间架起了一座桥梁，它允许我们使用复杂的代数理论来计算球面的同伦群。这座桥被称为色同伦理论。这个项目研究了这一领域中最重要的两个结构猜想，望远镜猜想和色分裂猜想。它有一个强大的计算组件，将提供数据来帮助研究这两个基本问题。色同伦理论使用了K-理论的更高相似性，从而在球面的稳定同伦群中产生更高的周期性。色分裂猜想是解释不同周期之间关系的一种尝试。最近，PI证明了这个猜想的一个特例。在与Goerss和Henn合作的一个项目中，PI旨在解释这一失败并重新表述猜想。在与Xu的一个相关项目中，PI计划计算K(2)-局部球面在素数2处的同伦群。这些计算使用似乎与Brown-Commenetz对偶和K(2)-局部Picard群的结构有关的自对偶分辨率。PI计划研究与Bobkova、Goerss和Henn的这种联系。该项目的一个平行部分是研究与望远镜猜想有关的周期性。在与Behrens、Bhattacharya、Culver和Xu的合作中，PI计划使用球面的拓扑模块形式的分辨率来检测周期元素。望远镜猜想给出了用这种方法探测到的周期元素与K(2)局域球面探测到的元素之间的联系。将这两个计算结果进行比较可能有助于解释这一猜想。

AbstractAward: DMS 1612020, Principal Investigator: Agnes BeaudryThe project supported by this grant is part of a large scale effort to understand the homotopy groups of spheres, one of the fundamental problems in algebraic topology. The spheres, which are among the simplest geometric objects, are building blocks for more complex entities. The homotopy groups of spheres, which are used to study the connections between these basic components, are collections of continuous functions between spheres considered up to deformations. Many problems in other fields, especially differential topology, have been reduced to the study of these groups, which is notoriously difficult. However, a bridge has been built between algebraic geometry and algebraic topology that allows us to use sophisticated algebraic theory to enable calculations of the homotopy groups of spheres. This bridge is known as chromatic homotopy theory. This project studies two of the most important structural conjectures in this field, the telescope and chromatic splitting conjectures. It has a strong computational component that will provide data to help study these two fundamental problems.Chromatic homotopy theory uses higher analogues of K-theory which give rise to higher periodicity in the stable homotopy groups of spheres. The chromatic splitting conjecture is an attempt at explaining the relationship between different periodicities. Recently, the PI has disproved a special case of this conjecture. In a project with Goerss and Henn, the PI aims to explain this failure and reformulate the conjecture. In a related project with Xu, the PI plans to compute the homotopy groups of the K(2)-local sphere at the prime two. These computations use self-dual resolutions which appear to be related to Brown-Commenetz duality and the structure of the K(2)-local Picard group. The PI plans to study this connection with Bobkova, Goerss and Henn. A parallel part of the project is to study periodicity in relation to the telescope conjecture. In work with Behrens, Bhattacharya, Culver and Xu, the PI plans to detect periodic elements using a resolution of the sphere by topological modular forms. The telescope conjecture gives a connection between periodic elements detected by such methods and those detected by the K(2)-local sphere. Comparing the two computations may shed light on this conjecture.