Park City Mathematics Institute
Curvature Group
Project Abstract
Our working group, the Curvature Crew, attended the Undergraduate Summer School lectures on the Curvature of Space and Time, given by Iva Stavrov, Associate Professor of Mathematics at Lewis & Clark College. In addition, we spend an hour after each lecture debriefing concepts and working on problem sets with Brian Hopkins, SSTP staff.

Two weeks of the course were devoted to Riemannian geometry and curvature, the last week focusing on general relativity, SpaceTime, and geometric analysis. More details can be found in the weekly summaries.

While much of the course was very abstract, Iva did a good job of giving examples and exercises in specific settings, such as polar and spherical coordinates, and various models of hyperbolic space. We proved, using some Christoffel symbols (certain partial derivatives), that great circles really are the geodesics on the sphere. In the upper half plane model of hyperbolic space, we found arclengths along semicircles, finally having a reason to integrate cosecant.

Our richest investigation was an exercise from Thurston about creating discrete 2-manifolds by taping together triangles. We found the tetrahedron, octahedron, and icosahedron (all positive curvature like the sphere); the plane (zero curvature); and a ruffly surface with seven triangles around each vertex (negative curvature like hyperbolic space). Exploring the number of triangles distance k from a vertex in each setting led to nice formulas and a surprising occurrence of the Fibonacci & Lucas numbers (see Week 3 summary).

We also appreciated how Iva used historical motivation throughout her lectures and notes. We started with a lecture by Riemann and eventually talked about a discrepancy concerning Mercury's orbit being resolved by Einstein's new theories.

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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.