Abstract: I will talk about estimating the length of a shortest closed geodesic on simply connected manifolds. I will present two different type of estimates. Results of the first type will be upper bounds for the length of a shortest closed geodesic on a closed Riemannian manifold of any topological type in terms of either a lower bound on sectional curvature, an upper bound on the diameter and a lower bound on the volume of a manifold, or in terms of an upper bound on sectional curvature and an upper bound on the volume. The technique involves finding an "optimal'' homotopy of contracting contractible spheres.
The second type estimates will be improved upper bounds for the length of a shortest closed geodesic $l(M)$, where $M$ is diffeomorphic to the two-dimensional sphere. We will show that $l(M) \leq 4d$ and $l(M) \leq 8 \sqrt{A}$, where $d$ is the diameter and $A$ is the area of $M$. The technique involves some geometric measure theory.