Local and Global Chromatic Stable Homotopy Theory

[摘要]获颁：DMS 1308916，首席研究员：Paul G. goerss稳定同伦的色图利用形式群的代数几何来组织和指导对该领域更深层次结构的研究。这项赠款支持的项目是收集当地信息——从单一高度的正式群体中可以看到的数据——然后将这些数据整合成更全面的图景。在第二步中，我们可以使用衍生代数几何的结构和信息；这允许我们在高度之间进行插值。本提案重点关注四个项目，它们都源于这种地方与全球的结合。最具计算性的是对K(2)局部球的同伦群的研究；这就是我们在高度2处看到的。这个长期的项目，与汉斯-维尔纳·亨恩和其他人；我们在低质数时看到了美丽而意想不到的现象。第二个，密切相关的项目，是研究Morava e -理论对Morava稳定群的某些封闭子群的不动点谱。这些比球体本身简单得多，但却包含了很多重要的同伦理论。另外两个项目在本质上更具全球性。一是研究具有水平结构的椭圆曲线的导出格式（或堆栈）的存在性和不存在性；也就是霍普金斯-米勒拓扑模形式理论的结构化版本。这里的要点是对等变结构作一个系统的研究。另一个项目是通过p可分群的透镜来观察色分裂猜想。所有这些项目都属于同伦理论，这是拓扑学的一个分支。这一领域的主要目的是研究在连续变换下保持不变的数学现象。许多熟悉的几何现象——比如角度——在这种情况下并不是不变的；然而，持续的转变是自然而丰富的。长期的研究表明，最有成果的不变现象之一是从圆，或者更一般地说，是从高维球体到待研究空间的映射。这些是同伦群。从历史上看，这些群体被描述为“碾磨人群”；然而，最近从数论和代数几何中引入的技术和结构使我们能够进行详细的计算，并发现具有显著规律性和美丽的大规模模式。

AbstractAward: DMS 1308916, Principal Investigator: Paul G. GoerssThe chromatic picture of stable homotopy uses the algebraic geometry of formal groups to organize and direct investigations into the deeper structures of the field. The program supported by this grant is to gather local information - the data that can been seen from formal groups of a single height - and then to assemble that data into a more global picture. It is in the second step where we can use constructions and information from derived algebraic geometry; these allow us to interpolate among heights. This proposal focuses on four projects, all growing out of this local-to-global mixture. The most computational is an investigation of the homotopy groups of the K(2)-local sphere; that is, what we can see at height 2. This long-standing project, with Hans-Werner Henn and others; we are seeing beautiful and unexpected phenomena at low primes. A second, closely related project, is to investigate the fixed point spectra of Morava E-theory for certain closed subgroups of the Morava stabilizer group. These are much simpler than the sphere itself, but capture a great deal of the important homotopy theory. The other two projects are more global in nature. One is to investigate the existence and non-existence of derived schemes (or stacks) elliptic curves with level structure; that is, structured versions of the Hopkins-Miller theory of topological modular forms. The point here is to make a systematic investigation of the equivariant structure. The other project is to look at the Chromatic Splitting Conjecture through the lens of p-divisible groups.All of these projects lie in homotopy theory, a branch of topology. The main aim of this field is to study mathematical phenomena which remain invariant under continuous transformations. Many familiar geometric phenomena - such as angles - are not invariant in this fashion; yet continuous transformations are natural and abundant. Long study has indicated that the among the most fruitful invariant phenomena are classes of maps from circles or, more generally, higher dimensional spheres, into the space to be studied. These are the homotopy groups. Historically these groups were described as a "milling crowd"; however, the recent introduction of techniques and constructions from number theory and algebraic geometry have permitted us to do detailed calculations and to uncover large scale patterns of remarkable regularity and beauty.