**What**: Topology Seminar**When**: 2:40 p.m. Wednesday, Feb. 1**Where**: Math/Geosciences Room 124**Speaker**: Jens Harlander

**Abstract**: If one tiles a sphere and orients the edges of the tiling then one of the following has to occur: there is a sink vertex, that is a vertex so that all its edges point toward it, or a source vertex, that is a vertex so that all its edges point away from it, or a consistently oriented region, that is a region whose boundary edges are all oriented the same way. This observation is due to John Stallings. One can prove it using the fact that the sphere is positively curved. In my talk I will introduce combinatorial curvature, prove Stallings’ observation and give some other applications of curvature techniques to problems in low dimensional topology.

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