Collaborative Research: FRG: Homotopical Approaches to Group Actions

DMS- 0354699dms -0354633 clarence W. Wilkerson, Jeffrey Smith, William Dwyer, Alexandro Adem， Jesper grodal这是DMS重点研究小组正在征求的奖项http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htm。主要研究人员是普渡大学的Clarence W. Wilkerson和Jeffrey Smith，圣母大学的William Dwyer，威斯康星大学的Alexandro Adem和芝加哥大学的Jesper Grodal。群体和他们的行为方式（换句话说，他们的表现）在数学中起着很大的作用。这个提议是关于研究拓扑空间上的群体行为。三种类型的群起作用：有限群、无限离散群和紧李群，它们的作用要么从几何角度考虑，要么从同调角度考虑。研究人员相信，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一些联系。从更广泛的角度来看，这个建议处理高维几何形状的对称性。给定一个特定的形状，要明确地确定其所有的对称性是非常困难的，但有时可以通过观察可能发生的对称类型如何受到形状的定性属性的影响来获得部分信息。例如，我们知道，一个有限维的无环形状（粗略地说，没有孔）不可能有一组有限的对称，这些对称的性质是每个单独的非恒等对称都会移动形状的每个点。最近的一些发现使这类定性问题更容易解决，研究人员希望利用这些发现来解决一些长期存在的问题，并获得对对称性性质的新的理论见解。

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.