Operads, Group Actions, and Classifying Spaces

DMS-0206963 Clarence W.WilkersonJames E.McClureJeffrey H.SmithWilkerson(与圣母大学的W.G.Dwyer合作)研究Lie群和p-紧群的分类空间，目的是完成2-紧群及其自同构群的分类。Wilkerson和Smith研究了有限群在任意有限复形上的作用。我们的目标是构造一个模空间，它用给定的定点数据来分类动作。McClure和Smith将继续他们在同伦理论的链算子和链模型方面的工作。具体地说，他们提出了框架小盘算子的一个小链模型，证明了某个链算子上的“不稳定”余代数范畴是HZ-局部同伦理论的一个模型，给出了一个类似的HZ-局部谱的模型，并为K(N)-模谱的模型范畴建立了一个链模型。他们还建议研究余单纯链复范畴上对称单形结构的性质。McClure利用与Smith的联合工作来研究Snaith分裂的同伦论性质。他希望找到Goerss-Hopkins定理的一个简化证明，该定理给出了谱E(N)的交换乘法。他还研究了群为圆时等变谱的有理同伦理论。Smith和Grodal正在研究同伦G-球面。也就是说，同伦等价于球面且具有有限群G的作用的空间，它们希望基于群的代数不变量给出一个完全分类。他们还研究了同伦G球模空间。同伦理论是所有几何中最基本的。它研究的是无论发生什么连续变形都不会改变的几何性质。甜甜圈和咖啡杯的“平等”就是一个众所周知的例子。然而，令人惊讶的是，同伦理论研究的几何具有内在的代数性质。PI使用来自代数的技术来研究空间的几何性质，同伦理论在这些不同的数学领域之间提供了桥梁。事实上，空间的所有同伦信息都可以用代数来描述。代数是复杂的，但同伦理论给出了几何和代数之间的对应关系，具有许多重要的应用。

DMS-0206963Clarence W. WilkersonJames E. McClureJeffrey H. SmithWilkerson (in joint work with W. G. Dwyer of Notre Dame) studies the classifying spaces of Lie groups and p-compact groups with the goal of finishing the classification of 2-compact groups and their automorphisms. Wilkerson and Smith study actions of finite groups on arbitrary finite complexes. The goal is to construct a moduli spacethat classifies actions with given fixed point data.McClure and Smith will continue their work on chain operads and chain models for homotopy theories. Specifically, they propose to find a small chain model for the framed little-disks operad, to show that the category of "unstable" coalgebras over a certain chain operad is a model for HZ-local homotopy theory of spaces, to give a similar model for HZ-local spectra, and to create a chain model for the model category of K(n)-module spectra. They also propose to investigate the properties of a symmetric monoidalstructure on the category of cosimplicial chain complexes.McClure uses the joint work with Smith to study the homotopytheoretic properties of the Snaith splitting. He hopes to find a simplified proof of the theorem of Goerss-Hopkins theorem which gives the spectrum E(n) a commutative multiplication. He also studies the rational homotopy theory of equivariant spectra when the group is the circle.Smith and Grodal are studying homotopy G-spheres. That is, spaces that are homotopy equivalent to a sphere and have an action of a finite group G. They hope to give a complete classification based on algebraic invariants of the group. They also study the moduli space of homotopy G-spheres.Homotopy theory is the most fundamental of all geometries. It studies those geometric properties which do not change no matter what continuous deformations are made. The "equality"of donuts and coffee cups is a well known example. Yet, surprisingly, geometry as studied by homotopy theory has an intrinsic algebraic nature. The PIs study the geometric properties of spaces using techniques that come from algebra, with homotopy theory providing the bridge between these different areas of mathematics. In fact, all homotopy information of a space can be described using algebra. The algebra is complicated but homotopy theory gives a correspondence between geometry and algebra that has many important applications.