Given a compact Riemannian manifold M, one can ask whether there is a closed geodesic lying on M and further, if so, how many? A long standing conjecture is that there are infinitely many closed geodesics. In this talk, I will explain how this problem divides up into a geometric and a homotopy-theoretic part and I will discuss progress on the homotopy-theoretic side.
Much of the talk will be elementary and accessible to graduate students.