Many thanks for 10k subscribers! Fun video for you from Topology: The Borsuk-Ulam Theorem. One interpretation of this is that on the surface of the earth, there must be some point where it and its antipode (the spot exactly opposite it) have the exact same temperature and pressure. More generally, if you have a continuous map f:S^n to R^n then there exists a point x where f(x) = f(-x).

We will see some animations to get a feel for these kinds of functions, prove it from elementary methods in dimension 1 using Intermediate Value Theorem, look at a very cool set-theoretic corolloray called the Lusternik–Schnirelmann and finally see how it connects to Brower's Fixed Point theorem and various lemma's in combinatorics.

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This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.

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