When can R^n be a division algebra? This is a very deep question answered by Adams(1960): when n=1,2,4 or 8. It's not obvious that this question is related to algebraic topology, and what plays a crucial role here is the Hopf invariant.
In my talk, I'll introduce the Hopf invariant and the solution to the above "Hopf invariant one problem". Then I'll introduce the Kervaire invariant. These two invariants look very different, but they have some connections in stable homotopy theory, and both play a vital part in the development of algebraic topology.