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

空间X的一组对称诱导X的某些代数不变量(所谓的同调表示)上的对称性。这样，变换群理论(研究连续物体的对称性，如空间)和表示理论(研究离散或代数物体的对称性)变得密切相关。这个项目的目标是系统地研究对称几何(如不动点)和它们的代数表现之间的相互作用。特别地，具有指定代数表示的Moore空间的对称性的存在性问题(在1960年代初S提出的Steenrod问题)、代数曲面和某些4维空间都提供了一般理论应该应用于其上的具体问题。来自拓扑学、代数几何学和表示论的技术应该在这一努力中进行卓有成效的互动。对称性在科学中发挥着绝对的基础作用(事实上，在艺术和音乐中也是如此)，我们试图通过定性和某些定量的测量和比较来理解这种作用。在数学中，对称性也扮演着基本的角色，它有许多不同但又相互关联的表现形式。为了对数学结构(如空间)的对称性有一个精确而合乎逻辑的理解，我们需要发现并量化它的所谓不变量和其他重要特征。多年来，在不同的研究项目中开发了这一行业的各种复杂工具，主要是代数工具，这位研究员擅长并计划在他目前的工作中利用数量惊人的工具。

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.