Chromatic Stable Homotopy Theory and Derived Algebraic Geometry

稳定同伦的色图利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。基本的计划是收集当地的信息，然后试图将这些数据组合成更全球化的图景。在第二步中，我们可以使用派生代数几何这一新兴领域的构造和信息。这项提案集中在三个项目上，都是在这种从本地到全球的混合体中发展起来的。计算量最大的是研究$K(2)$-局部球面的同伦群；这本质上是局部的，我们寻求一个完整的计算。另外两个项目更具全球性。第一个是研究具有层次结构的派生方案(或栈)的存在性和不存在性，即Hopkins-Miller拓扑模形式的结构化形式。这里有趣的是从超奇异曲线产生有趣的同伦理论的糟糕的素数。这里的另一个项目是研究二元性。Serre-Grothendieck对偶的一种形式应该在派生的背景下成立，但它本质上是同伦理论，而不是简单的代数几何。这个项目是在同伦理论中进行的，同伦理论是拓扑学的一个分支，通过研究在连续变换下保持不变的现象，而不是刚性的(例如，保角的)变换，几何自然地发展出一个现代领域。在拓扑学中特别重要的是高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。这是出了名的难以计算，甚至难以猜测；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。使用其他领域，特别是代数几何的工具来检测这些现象是非常有成效的。从拓扑到几何的转换是使用同调理论完成的，同调理论是将拓扑学中的行为线性化的一种方法。然而，简单地坚持一个这样的理论是一个激进的过程，它丢失了太多的数据；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，因为它允许我们研究几何对象的连续族之间的对称性--特别是当自对称性在整个族中可能不连续地变化时，就像这里最肯定的情况。

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.