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.

稳定同构的色彩图片使用形式组的代数几何形状来组织和直接研究该领域的更深层结构。基本程序是收集本地信息，然后尝试将这些数据组装成更全球的图像。在第二步中，我们可以使用来自派生的代数几何形状的新兴领域的构造和信息。 该提案的重点是三个项目，所有项目都从这种局部到全球混合物中发展出来。最多的计算是对$ k（2）$ - 本地领域的同拷贝组的调查；从本质上讲，这是本地的，我们寻求完整的计算。 其他两个项目更全球。首先是研究具有级别结构的衍生方案（或堆栈）的存在和不存在。也就是说，霍普金斯 - 米勒拓扑模块化形式的结构化版本。这是不良的素数，其中有趣的同义理论是由超级曲线引起的。这里的另一个项目是二元性。一种形式的形式在派生的环境中应该存在，但是它本质上是同质理论的，而不仅仅是代数几何。该项目是同型理论，它是拓扑结构的一个分支，这是一个现代领域，这种现代领域自然地逐渐消失了几何形状，而不是在持续的变换中，而不是在持续的变换下，而不是越来越多的变换。拓扑中尤为重要的是大维球体之间的连续图。在适当的等效关系下，这是稳定的球体稳定均值组的环。这个臭名昭著的难以计算，甚至是对猜想做出的；因此，在过去的几十年中，我们专注于试图了解大规模的定性现象。总而言之，这也是该项目的主要目的。使用来自其他字段的工具，尤其是代数几何形状来检测这些现象是非常富有成效的。从拓扑到几何学的过渡是使用同源性理论完成的，这是拓扑中线性化行为的一种方式。但是，仅仅坚持这样一个理论就是一个根本的过程，它失去了太多数据。因此，我们研究了这种理论的家庭。 堆栈的理论在这里至关重要，因为这使我们能够研究几何学对象的连续家族之间的对称性 - 尤其是当自我对称性在整个家庭中都可能不断变化时，就像这里的情况一样。