Curvature Group Summary

Monday - Friday, July 8 - 12, 2013

During the second week of "The Curvature of Space and Time" lectures, we finally developed enough calculus to talk sensibly about curvature of non-Euclidean spaces. A way to think about the curvature of space is by considering the areas and circumferences of "circles" in our space, compared to circles in the Euclidean plane. A space with positive curvature (like spheres) corresponds to areas and circumferences less than Euclidean circles, while space with negative curvature (like hyperbolic space) correspond to areas and circumferences greater than Euclidean circles.

We continued to meet for an hour after each lecture to debrief and solve problems. On Friday, we built a model of the sphere and hyperbolic space using triangles (and lots of tape). This allowed us to visualize the deviation from the usual Euclidean area. Because the area of each "circle" is smaller, the shape eventually folds up and becomes sphere-like. In hyperbolic space, each "circle" is larger, and in 3-space looks ruffly.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.