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.

稳定同伦的色图像使用形式群的代数几何来组织和指导对场的更深层次结构的研究。基本程序是收集本地信息，然后尝试将这些数据组合成一个更全局的图像。这是在第二步，我们可以使用的结构和信息，从新兴领域的派生代数几何。 这一提议侧重于三个项目，都是从这种地方到全球的混合中发展出来的。最计算的是调查同伦群的$K（2）$-本地球;这是本地的性质，我们寻求一个完整的计算。 另外两个项目更具有全球性。第一个是调查存在和不存在的衍生计划（或堆栈）的水平结构，即结构化版本的霍普金-米勒拓扑模形式。有趣的是这里的坏素数有趣的同伦理论产生于超奇异曲线。这里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导出的设置中保持不变，但它本质上是同伦理论，而不仅仅是代数几何。这个项目是同伦理论，它是拓扑学的一个分支，一个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球面之间的连续映射;在适当的等价关系下，这是球面的稳定同伦群的环。这是众所周知的难以计算，甚至难以解释的;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。总而言之，这也是这个项目的主旨。利用其他领域的工具，特别是代数几何工具来探测这些现象，已经取得了丰硕的成果。从拓扑学到几何学的过渡是使用同调理论完成的，这是一种将拓扑学中的行为线性化的方法。然而，简单地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，我们研究这样的理论家族。 堆栈理论在这里是至关重要的，因为这使我们能够研究几何对象的连续族的对称性-特别是当自对称性可以在整个族中非连续地变化时，就像这里的情况一样。