Collaborative Research: FRG: Homotopical Approaches to Group Actions

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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.