Mathematical Sciences: Topological Methods in Algebraic Transformation Groups

这个项目从光滑(可微)变换群的观点来处理仿射簇上的代数(和解析)作用。光滑变换群的研究受到以下两个核心问题的强烈影响：线性问题。哪些群在某一维的仿射空间上起到了非线性作用？(如果一个动作与由群的表示定义的动作共轭，则该动作是线性的。)不动点问题。哪些群在某一维仿射空间上没有不动点？用来解决这些问题的工具以及它们所产生的进一步问题一直是光滑变换组领域的重要主题。在过去的几年里，来自代数变换群和光滑变换群的研究人员已经认识到这两个问题在代数范畴中的重要性，在这种情况下，所涉及的群应该是约化的(作为特例包括有限群)。皮特里将在这个方向上继续他的工作，将拓扑学方法应用于代数问题。

This project treats algebraic (and analytic) actions on affine varieties from the viewpoint of smooth (differentiable) transformation groups. The subject of smooth transformation groups has been strongly influenced by the following two central problems: LINEARITY PROBLEM. Which groups act nonlinearly on affine space of some dimension? (An action is linear if it is conjugate to an action defined by a representation of the group.) FIXED POINT PROBLEM. Which groups act without fixed points on affine space of some dimension? The tools used to settle these problems and the further problems they generated have been important themes in the field of smooth transformation groups. During the last few years researchers from algebraic transformation groups as well as those from smooth transformation groups have recognized the importance of these two questions in the algebraic category, and in that case the groups dealt with should be reductive (which includes as a special case the finite groups). Petrie will continue his work in this direction, bringing topological methods to bear on algebraic questions.