The previous talk in the series is here. The next talk in the series is here.
The Borsuk-Ulam theorem states that if f is a continuous function from Sn to ℝn (that is, from the n-sphere to n-dimensional Euclidean space), then there exists x such that f(x) = f(-x). It has many applications, including in combinatorics. In this video I prepare the ground for explaining two of these applications by discussing various statements that are equivalent to the theorem. We introduce and state the Borsuk-Ulam theorem, then consider the closed-sets version and how it follows from the Borsuk-Ulam theorem. Afterwards, the closed-sets version implies the Borsuk-Ulam theorem, then the closed-sets version implies the open-sets version, and finally the open-sets version implies a ‘mixed’ version and hence the closed-sets version.
This video was produced by Tim Gowers as part of his Part III course at the University of Cambridge. Printed notes for this course are available here.