Operads, Group Actions, and Classifying Spaces

jeffrey H. SmithWilkerson（与Notre Dame的W. G. Dwyer合作）研究了李群和p紧群的分类空间，目的是完成2紧群及其自同构的分类。Wilkerson和Smith研究了有限群在任意有限复上的作用。目标是构造一个模空间，对给定不动点数据的动作进行分类。麦克卢尔和史密斯将继续研究同伦理论的链操作符和链模型。具体而言，他们提出寻找框架小盘算子的小链模型，证明某链算子上的“不稳定”余代数范畴是空间的hz -局部同伦理论的模型，给出了hz -局部谱的类似模型，并为K(n)-模谱的模型范畴建立了链模型。他们还提出在共链配合物的范畴上研究对称单一型结构的性质。麦克卢尔利用与史密斯的合作研究了Snaith分裂的同伦性质。他希望找到高斯-霍普金斯定理的简化证明，该定理给出了谱E(n)的交换乘法。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。即同伦等价于球且具有有限群g作用的空间，他们希望给出基于群的代数不变量的完全分类。他们还研究了同伦g球的模空间。同伦理论是所有几何中最基本的理论。它研究的是在任何连续变形下都不改变的几何性质。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，由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，同伦理论为这些不同的数学领域提供了桥梁。事实上，一个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.