# Mathematical Sciences

## Mark Walsh, Oregon State University

Date: 3:30pm PDT September 20, 2011 Location: 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.