Local and Global Chromatic Stable Homotopy Theory

局部和全局色稳定同伦理论

• 批准号: 1308916
• 负责人: Paul Goerss
• 金额: $ 29万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308916&HistoricalAwards=false
• 关键词: Local; Global; Chromatic; Stable; Homotopy

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.

奖：DMS 1308916，首席研究员：Paul G.Goers稳定同伦的色图使用形式群的代数几何来组织和指导对域的更深层次结构的研究。这笔赠款支持的计划是收集当地信息--可以从单一高度的正式群体中看到的数据--然后将这些数据汇编成更全球化的图景。在第二步中，我们可以使用派生的代数几何中的构造和信息；这些允许我们在高度之间进行内插。这项提案重点关注四个项目，它们都是在这种从本地到全球的混合体中成长起来的。计算量最大的是对K(2)-局部球的同伦群的研究；也就是我们在高度2可以看到的东西。这个长期的项目，与Hans-Werner Henn和其他人一起；我们在低质数看到了美丽和意想不到的现象。第二个密切相关的项目是研究Morava稳定子群的某些闭子群的Morava E-理论的不动点谱。它们比球体本身简单得多，但捕捉到了大量重要的同伦理论。另外两个项目在性质上更具全球性。一个是研究具有水平结构的派生格式(或堆栈)椭圆曲线的存在性和不存在性，即拓扑模形式的Hopkins-Miller理论的结构化版本。这里的重点是对等变结构进行系统的研究。另一个项目是通过p-可除群的透镜来研究色分裂猜想，所有这些项目都是在拓扑学的一个分支同伦理论中进行的。这个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--例如角度--并不是以这种方式不变的；然而，连续的变换是自然和丰富的。长期的研究表明，最有成果的不变现象之一是从圆或更广泛地说，高维球面到待研究空间的映射类。这些是同伦群。在历史上，这些群体被描述为“磨合的人群”；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感的大规模模式。