The determination of how many independent vector fields there are on the sphere was one of the great successes of algebraic topology.

The story starts with the Hairy Ball Theorem: if is even, there are no nowhere zero (continuous) vector fields on . We must then turn to odd dimensional spheres, and wonder how many linearly independent vector fields can there be? On our way to answering this question, we will encounter many familiar structures: , , and even more general algebraic structures known as Clifford algebras, which will allow us to give a lower bound on the number of linearly independent vecor fields on spheres.

We will find many relations with K-Theory, similarities and common periodicities; these have been exploited by Frank Adams to prove that the lower bound provided by Clifford algebras is actually an equality, although we will not delve into the details.

Come tonight at MS.05 to hear about all this, and more! After which we find our way to the pub.