Mathematics Colloquia and Seminars

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For each Riemannian metric on a surface S, one can consider the shortest geodesics in each homotopy class of curves in S, and how these geodesics change when the surface is deformed by changing the Riemannian metric. I will review some known results and present new ones (in collaboration with Peter Scott) that include bounds for the number of different configurations in each homotopy class, and for hyperbolic metrics, bounds for the angles of intersection that depend only on the lengths of the geodesics. I will also consider the problem of characterizing all the possible "length functions" (that give the lengths of the shortest geodesics in each homotopy class) for all metrics on S, and how they relate to the the geometric intersection numbers between all homotopy classes of curves in S.