The Borsuk--Ulam theorem, its generalizations, and applications

The Borsuk--Ulam theorem has found numerous applications across mathematical disciplines since its discovery in the 1930s. The theorem states that any continuous map from a d-sphere to d-space identifies a pair of antipodal points. I will show that this result remains relevant today and present new consequences. I will present two recent generalizations of the Borsuk--Ulam theorem, a colorful extension and a version for high-dimensional codomains, and explain some connections with packings of projective space among other topics.

This is joint work with Henry Adams, Johnathan Bush, and Zoe Wellner.