Event Type:

Department Colloquium

Date/Time:

Monday, November 11, 2013 - 08:00

Location:

Kidder 350

Guest Speaker:

Jens Harlander

Institution:

Boise State University

Abstract:

Euler's formula, published in 1758, states that if G is a planar connected graph then a_{0}-a_{1}+a_{2}=2, where a_{0} is the number of vertices, a_{1} is the number of edges, and a_{2} is the number of regions. At the beginning of the 20th century Poincare observed (but did not prove) that Euler's formula holds true in a much more general setting. If a space X is divided up into finitely many cells, then the alternating sum over n of the number of n-cells, does not depend on the division. This alternating sum is referred to as the Euler characteristic c(X). The Gauss-Bonnet theorem (known to Gauss but proved by Bonnet in 1848) provides a differential geometric version of Euler's formula. It states that the total curvature of a surface M is 2p c(M). Intuitively this says that the total curvature of a surface embedded in 3-space is independent of the embedding. Of central importance in combinatorial topology is a combinatorial version of the Gauss-Bonnet theorem. In my talk I will state and prove this theorem and will survey applications to group theory and 2-dimensional topology.

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