Gracious Living

Differential Geometry and the Sphere Theorem
December 16, 2010, 20:38
Filed under: Topology, Uncategorized | Tags: , , , ,

I just wrote a paper about the Sphere Theorem for my differential geometry class.  Since I can’t really get it out of my head, I thought it’d be fun to use to give a brief overview of differential/Riemannian geometry, in more or less layman’s terms.  This is somewhat of a departure from my normal posting style: though you’ll get more out of this if you’ve understood the topology posts so far, I’ll try to write in broad enough strokes that calculus is the only prerequisite.  Really, the most complicated idea behind this is the idea of a map from $\mathbb{R}^m$ to $\mathbb{R}^n$ being differentiable.  I also refer to vectors, but I think that that’s a pretty intuitive concept, in general.

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Local Connectedness and Some Weird Counterexamples
November 10, 2010, 14:25
Filed under: Math, Topology | Tags: , , , , ,

Last time we talked about connectedness and path-connectedness and showed that though all path-connected spaces are also connected, connected spaces are not in general path-connected. There is a partial remedy to this problem, found in the local versions of the connectedness properties.  Recall that a local property says something about neighborhoods of each point, rather than the space itself or its open sets.  Below the fold, I talk about local connectedness and local path-connectedness, and bring up a couple examples and counterexamples, including one of my favorites, the Conway base-13 function.

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