Collaborative Research: FRG: Homotopical Approaches to Group Actions

DMS-0354787DMS-0354699DMS-0354633Clarence W.威尔克森，杰弗里史密斯，威廉德怀尔，亚历山大阿德姆，杰斯珀格罗达尔这是一个DMS重点研究小组奖下征求http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htm。 主要研究者是Clarence W.普渡大学的威尔克森和杰弗里·史密斯，圣母大学的威廉·德怀尔，威斯康星州大学的亚历山大·阿德姆和芝加哥大学的杰斯珀·格罗达尔。 群和它们的行为方式（换句话说，它们的表示）在数学中起着很大的作用。这个建议是关于研究拓扑空间上的群作用。 三种类型的群起作用：有限群，无限离散群和紧李群，它们的作用被认为是从几何或同伦的角度。 研究者相信他们可以利用同伦理论的最新进展来获得关于这些群的新信息，特别是澄清群的内部结构和它可以作用的空间的几何性质之间的联系。 给定一个特定的形状，很难明确地确定它的所有对称性，但有时可以通过观察形状的定性属性如何影响可能发生的对称性类型来获得部分信息。例如，已知非循环的有限维形状（粗略地说，没有孔）不可能有一个有限的对称群，其性质是每个单独的非单位对称移动形状的每个点。最近的一些发现使这类定性问题的解决变得更加容易，研究人员希望利用这些发现来解决一些长期存在的问题，并获得对对称性的新的理论见解。

DMS-0354787DMS-0354699DMS-0354633Clarence W. Wilkerson, Jeffrey Smith, William Dwyer, Alexandro Adem, Jesper GrodalThis is a DMS Focused Reseach Group award under solicitation http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htm. The principal investigators are Clarence W. Wilkerson and Jeffrey Smith at Purdue University, William Dwyer at the University of Notre Dame University, Alexandro Adem at the University of Wisconsin, and Jesper Grodal at the University of Chicago. Groups and the ways in which they act (in other words, their representations) play a large part in mathematics. This proposal is concerned with studying group actions on topological spaces. Three types of groups play a role: finite groups, infinite discrete groups, and compact Lie groups, and their actions are considered either from a geometrical or a homotopical point of view. The investigators believe that they can leverage recent advances in homotopy theory in order to obtain new information about these groups, and in particular to clarify some ties between the internal structure of a group and the geometrical properties of a spaces on which it can act.From a broader point of view, this proposal deals with symmetries of higher-dimensional geometrical shapes. Given a particular shape, it can be very difficult to determine all of its symmetries explicitly, but sometimes it is possible to obtain partial information by looking at how the types of symmetries that can occur are affected by qualitative properties of the shape. For instance, it is known that a finite-dimensional shape that is acyclic (roughly, has no holes) cannot have a finite group of symmetries with the property that each individual non-identity symmetry moves every point of the shape. Some recent discoveries make it easier to approach qualitative questions of this kind, and the investigators hope to use these discoveries to solve some longstanding problems and to gain new theoretical insight into properties of symmetries.