Topological and Geometric Rigidity James F. Davis (Indiana University), Shmuel Weinberger (University of Chicago) July 29 – August 3, 2007 1 Introduction This conference discussed a wide variety of problems motivated directly and indirectly by Mostow rigidity and a wide variety of approaches to these problems using tools from geometry, algebra, quantitative topology, and index theory. We shall give here a brief survey of these problems by mentioning at the appropriate places where talks in this conference fit in providing context for the work of the conference and an indication of that work. In particular, these notes are variants of the talks that Davis and Weinberger gave at the meeting. At the end of each section we will give a few references for additional information. We will often ref- erence a survey rather than an original source. Also, our mention of the topic discussed by a speaker is not intended to indicate that it was not an exposition of joint work. 2 Mostow rigidity (an example of geometric rigidity) It appropriate to begin our survey with Mostow Rigidity: Theorem 1 Suppose M and N are closed hyperbolic manifolds of dimension n> 2, and f : π 1 M → π 1 N is an isomorphism, then there is a unique isometry F : M → N , inducing f . For n =2 this is false, with there being a contractible space of hyperbolic structures on any surface, Teichmuller space. The theorem is true for noncompact manifolds with finite volume (Prasad) and for irre- ducible lattices in semisimple Lie groups (Mostow). Later we will discuss Margulis’s superrigidity which is a far-reaching extension of Mostow’s theorem. Without semisimplicity, isometry is too much to hope for: indeed even the torus has a large, yet con- tractible, space of flat structures SL n /SO n . Nevertheless, even this can be thought of as a form of rigidity (and the nonpositive curvature of that space does indeed give rise to a number of important applications.) Belegradek’s talk at the conference, among other things, give more examples of these kinds of classical rigidity phenomena. References for this section include: [31] and [34]. 3 Borel conjecture (an example of topological rigidity) Borel, in a letter to Serre, after hearing Mostow talk about rigidity (actually in the solvable setting where a smooth, rather than geometric rigidity, occurred) made the following far-reaching conjecture: 1