Singularities in Geometry and Topology

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AbstractAward: DMS-0706968Principal Investigator: Jeff Cheeger, Shmuel WeinbergerThis conference will explore differing approaches to non-smoothspaces that arise in modern geometry and topology with an eyetowards cross-fertilization among the various groups ofresearchers. Two of the central thrusts are topological methodsbased on the ideas of a stratification, and metric measurespaces, and their analysis, especially their differentialcalculus. Others are based on cohomology with growth conditions,or resolution of singularities, intersection homology, sheaftheory etc. Moreover the applications of these studies arediverse, from the Novikov conjecture, knot theory, andtransformation groups to Gromov-Hasudorff convergence techniquesin Riemannian geometry and geometric group theory to problems ofinterest to theoretical computer scientists (embedding finitemetric spaces in various Banach spaces). It is anticipated thatas a result of this meeting, progress can be expected in thedirections of producing analytic geometric structures on singularspaces, the quasi-isometric theory of groups, and characteristicclasses.Many of the basic ideas of analysis and geometry are based on theidea of linear approximation. The first example of this iscalculus itself where linear approximation defines the basicnotion of the derivative. A more modern example is the idea of amanifold in topology, which is a space that is, in small regions,well approximated by Euclidean space. Countless furtherdevelopments have shown the need for more general tools to dealwith situations where such approximations are not feasible or maynot exist. By bringing together workers who have dealt withthese issues in varied contexts, this conference will spur thedevelopment of new syntheses and bring to the fore common issuesthat arise in completely different settings. The conference website is http://www.math.uchicago.edu/~shmuel/Cappelliday.