Operads, Group Actions, and Classifying Spaces

DMS-0206963Clarence W.威尔克森Jeffrey H. SmithWilkerson（与W. G. Dwyer）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。 威尔克森和史密斯研究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，对给定不动点数据的行为进行分类。麦克卢尔和史密斯将继续他们在同伦理论的链运算和链模型方面的工作。 具体地说，他们提出为框架小圆盘操作寻找一个小链模型，证明在一定链操作上的“不稳定”余代数范畴是HZ-局部同伦空间理论的一个模型，给出HZ-局部谱的一个类似模型，并为K（n）-模谱的模型范畴建立一个链模型。 麦克卢尔利用与Smith的联合工作研究了Snaith分裂的同伦性质。 他希望找到一个简化的证明定理的高斯-霍普金斯定理，使频谱E（n）的交换乘法。他还研究了有理同伦理论的等变谱时，该集团是圆。史密斯和格罗达尔正在研究同伦G球。也就是说，空间同伦等价于一个球，并有一个有限群G的作用。他们希望根据群的代数不变量给出一个完整的分类。 同伦理论是所有几何学中最基本的理论。它研究的是那些无论如何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学具有内在的代数性质。 PI使用来自代数的技术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，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.