Formal Groups, Structured Ring Spectra, and Stable Homotopy Theory

摘要奖：DMS-0706705主要研究人员：保罗·G·戈尔斯稳定同伦的色图利用形式群的代数几何来组织和指导对计算和理论的更深层次结构的研究。这个项目寻求在两个方向上发展这一观点，一个是局部的，一个是全球的。第一个或局部方向是对一般K(N)-局部同伦理论的研究，特别是对K(2)-局部球面的研究。第二个或更具全局性的方向是，使用形式基团的堆栈和模堆栈作为基本的参数化工具，使我们的结构化环谱知识系统化。特别是，该项目的一个主要部分是继续研究在形式群的模组上实现交换环谱族的问题。Lurie，Behrens和Lawson最近的工作以及Lurie，Behrens和Lawson最近的工作都给出了拓扑模的谱形式。我们可以要求沿着这些路线获得系统的结果，我们可以要求对我们拥有的例子进行彻底的调查。这些模型的一个有趣而新颖的特点是，它们使用BarsottiTate群的理论来组合来自不同高度形式群的信息。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，拓扑学是一个相当现代的领域，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，自然地从几何中发展出来。在拓扑学中特别重要的是大维度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。使用其他领域的工具，特别是代数几何的工具来检测这些现象是非常有成效的。从拓扑学到几何的转换是利用同调理论来完成的，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繁琐的过程，它丢失了太多的数据；因此，我们研究这样的理论家族。特别重要的是由单参数形式Lieggroup的堆栈参数化的族。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，就像这里最确定的情况。

AbstractAward: DMS-0706705Principal Investigator: Paul G. GoerssThe chromatic picture of stable homotopy uses the algebraicgeometry of formal groups to organize and direct investigationsinto the deeper structure of computations and theory. Thisproject seeks to develop this point of view in two directions,one local and one global. The first, or local, direction is aninvestigation into K(n)-local homotopy theory in general and intothe K(2)-local sphere in particular. The second, or more global,direction, would be to make systematic our knowledge ofstructured ring spectra using stacks and the moduli stack offormal groups as the basic parameterizing device. In particular,a main part of the project is to continue work on the problem ofrealizing families of commutative ring spectra over the modulistack of formal groups. The spectrum of topological modular formsarises from taking the homotopy inverse limit of just such afamily and recent work of Lurie, Behrens, and Lawson had givennew examples. We can ask for systematic results along theselines, and we can ask for a thorough investigation into theexamples we have. An intriguing and novel feature of thesefamilies is that they use the theory of Barsotti-Tate groups tocombine information from formal groups of various heights.This project is in homotopy theory, which is a branch oftopology, a rather modern field that grew naturally out ofgeometry by studying phenomena that remain invariant undercontinuous transformations, rather than rigid (e.g.,angle-preserving) transformations. Of particular importance intopology are the continuous maps between large dimensionalspheres; under a suitable equivalence relation, this is the ringof stable homotopy groups of spheres. This notorious difficult tocalculate, or even to make conjectures about; therefore, in thepast few decades we have focused on trying to understandlarge-scale qualitative phenomena. In summary, this is the mainthrust of this project as well. It has been very fruitful todetect these phenomena using tools from other fields, especiallyalgebraic geometry. The transition from topology to geometry isdone using homology theories, which is a way of linearizingbehavior in topology. Simply sticking to one such theory is aradical process, however, and it loses too much data; therefore,we study families of such theories. Of particular importance isthe family parametrized by the stack of one-parameter formal Liegroups. The theory of stacks is vital here, as this allows us tostudy symmetries across continuous families of geometric objects-- especially when the self-symmetries can vary non-continuouslythroughout the family, as is most certainly the case here.