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Mathematical Sciences

Mark Walsh, Oregon State University

Date: 3:30pm PDT September 20, 2011 Location: Howard Hall, Room 244

Howard Hall, Room 244

Finding the Right Geometry

An important theme in modern geometry concerns the relationship between curvature and topology. Curvature is something which is fairly intuitive. A round sphere is positively curved, a sheet of paper is flat (the familiar geometry of Euclid), while a saddle surface displays negative curvature. Topology deals with the aspects of a shaped which are preserved under continuous deformation. Thus, a basketball is topologically equivalent to a football (as one could imagine a continuous deformation from one to the other) but neither is topologically equivalent to the surface of a bagel.

Given a particular shape one may study how its curvature changes under continuous deformation. A classical problem therefore, is to find the best kind of curvature or “geometry” for this topological type. For example, a torus (the surface of a bagel) cannot have curvature which is positive everywhere. However it is possible to make it flat (although one needs to be in a space of at least four dimensions to see this happen).

In this talk, the relationship between curvature and topology will be discussed with a particular emphasis on dimension two as this is the dimension we understand best, as well as a bit about the analogous problems in higher dimensions.