Mathematical Sciences: Geometric and Cohomological Methods in Transformation Groups and Representation Theory

空间X的一组对称诱导了 X的某些代数不变量（所谓的同调 表示）。 变换群理论（transformation group theory） 研究连续物体（如空间）的对称性， 表示论（研究离散或非离散系统的对称性， 代数对象）变得密切相关。 这个目标 该项目旨在系统地研究 对称的几何（如不动点）及其 代数表现 特别是存在的问题 摩尔空间的对称性 代表（斯廷罗德问题，提出于20世纪60年代初）， 代数曲面和某些四维空间都提供了 一般理论应该适用的具体问题。 拓扑学、代数几何学和表示法的技巧 理论应该在这一奋进中卓有成效地相互作用。 对称性在科学中起着绝对的基础作用（在 事实上，在艺术和音乐中也是如此），我们试图理解这一点， 通过定性和某些定量衡量， 比较。 在数学中，对称性起着基础性的作用 还有，它有很多不同的，但是相关的表现。 到 有一个精确的和逻辑的理解对称的一个 数学结构，如空间，我们需要发现和 量化其所谓的不变量和其他重要特征。 一 这一行业的各种复杂工具，主要是 代数的，已经发展多年，在不同的 研究项目，这位调查员擅长并计划 在他目前的工作中利用了数量惊人的这些理论。

A group of symmetries of a space X induces symmetries on certain algebraic invariants of X (so-called homology representations). In this way, transformation group theory (the study of the symmetries of continuous objects, such as spaces) and representation theory (the study of symmetries of discrete or algebraic objects) become intimately related. The goal of this project is to investigate systematically the interactions between the geometry of symmetries (such as fixed points) and their algebraic manifestations. In particular, the problem of existence of symmetries of Moore spaces with prescribed algebraic representations (the Steenrod Problem, posed in the early 1960's), algebraic surfaces, and certain 4-dimensional spaces all provide concrete problems to which a general theory should apply. Techniques from topology, algebraic geometry, and representation theory should interact fruitfully in this endeavor. Symmetry plays an absolutely fundamental role in science (in fact, in art and music as well), and we attempt to understand this role through qualitative and certain quantitative measurements and comparisons. In mathematics, symmetry plays a fundamental role also, and it has many different but related manifestations. To have a precise and logical understanding of symmetries of a mathematical structure, such as a space, we need to discover and quantify its so-called invariants and other important features. A wide variety of sophisticated tools of this trade, chiefly algebraic ones, have been developed over the years in different research projects, and this investigator is adept at and plans to draw upon a surprising number of them in his current work.