Colloborative Research: FRG: Homotopical Methods to Group Actions

DMS-0354787DMS-0354699DMS-0354633Clarence W.Wilkerson，Jeffrey Smith，William Dwyer，Alexandro Adem，Jesper Grodal这是一个DMS重点研究小组的奖项，由Solite http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htm.颁发主要调查人员是普渡大学的克拉伦斯·W·威尔克森和杰弗里·史密斯，圣母大学的威廉·德怀尔，威斯康星大学的亚历山德罗·阿登和芝加哥大学的杰斯珀·格罗达尔。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。这一建议涉及到研究拓扑空间上的群作用。有三种类型的群起作用：有限群、无限离散群和紧李群，它们的作用可以从几何或同伦的角度来考虑。研究人员认为，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，这一建议涉及更高维几何形状的对称性。给定特定形状，可能很难明确确定其所有对称性，但有时可以通过查看形状的定性属性如何影响可能出现的对称性类型来获得部分信息。例如，众所周知，一个非循环的有限维形状(粗略地说，没有洞)不可能有一个有限的对称组，其性质是每个单独的非恒等对称移动该形状的每个点。最近的一些发现使得研究这类定性问题变得更容易，研究人员希望利用这些发现来解决一些长期存在的问题，并对对称的性质获得新的理论见解。

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.