Chromatic Stable Homotopy Theory and Derived Algebraic Geometry

色稳定同伦理论及其派生代数几何

• 批准号: 1007007
• 负责人: Paul Goerss
• 金额: $ 28.77万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007007&HistoricalAwards=false
• 关键词: Chromatic; Stable; Homotopy; Theory; Derived

The 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 basic program is to gather local information and then try to assemble that data into a more global picture. It is in the second step where we can use constructions and information from the emerging field of derived algebraic geometry. This proposal focuses on three 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; this is local by nature and we seek a complete calculation. The other two projects are more global. The first is to investigate the existence and non-existence of derived schemes (or stacks) with level structure; that is, structured versions of the Hopkins-Miller topological modular forms. Of interest here are the bad primes where interesting homotopy theory arises from supersingular curves. The other project here is a look at duality. A form of Serre-Grothendieck duality should hold in the derived setting, but it will be homotopy theoretic in nature, not simply algebraic geometry.This project is in homotopy theory, which is a branch of topology, a modern field that grew naturally out of geometry by studying phenomena that remain invariant under continuous transformations, rather than rigid (e.g., angle-preserving) transformations. Of particular importance in topology are the continuous maps between large dimensional spheres; under a suitable equivalence relation, this is the ring of stable homotopy groups of spheres. This notorious difficult to calculate, or even to make conjectures about; therefore, in the past few decades we have focused on trying to understand large-scale qualitative phenomena. In summary, this is the main thrust of this project as well. It has been very fruitful to detect these phenomena using tools from other fields, especially algebraic geometry. The transition from topology to geometry is done using homology theories, which is a way of linearizing behavior in topology. Simply sticking to one such theory is a radical process, however, and it loses too much data; therefore, we study families of such theories. The theory of stacks is vital here, as this allows us to study symmetries across continuous families of geometric objects -- especially when the self-symmetries can vary non-continuously throughout the family, as is most certainly the case here.

稳定同伦的色图使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。基本的程序是收集当地的信息，然后试图将这些数据整合成一个更全面的图景。在第二步中，我们可以使用派生代数几何这个新兴领域的结构和信息。本提案聚焦于三个项目，它们都源于这种本地与全球的结合。最具计算性的是研究$K(2)$-局部球的同伦群；这是局部性质的，我们寻求一个完整的计算。另外两个项目更加全球化。首先是研究具有水平结构的衍生方案（或堆栈）的存在性和不存在性；也就是霍普金斯-米勒拓扑模形式的结构化版本。这里有一些坏素数有趣的同伦理论是从超奇异曲线中产生的。另一个项目是关于二元性的。Serre-Grothendieck对偶的一种形式应该在推导的背景下成立，但它本质上是同伦理论的，而不是简单的代数几何。这个项目属于同伦理论，它是拓扑学的一个分支，一个自然从几何学中发展出来的现代领域，通过研究在连续变换下保持不变的现象，而不是刚性（例如，保角）变换。在拓扑学中特别重要的是大维度球体之间的连续映射；在适当的等价关系下，这是稳定同伦群的环。这是出了名的难以计算，甚至难以猜测的；因此，在过去的几十年里，我们一直专注于试图理解大规模的定性现象。总之，这也是这个项目的主旨。利用其他领域的工具，特别是代数几何的工具来检测这些现象是非常富有成效的。从拓扑到几何的转换使用了同调理论，这是拓扑线性化行为的一种方法。然而，仅仅坚持一个这样的理论是一个激进的过程，它会丢失太多的数据；因此，我们研究这些理论的家族。堆栈理论在这里是至关重要的，因为它允许我们研究连续几何对象族之间的对称性——特别是当自对称性在整个族中可以非连续地变化时，就像这里的情况一样。