Geometry of 2-dimensional Riemannian disks and spheres.

I will discuss some geometric inequalities that hold on Riemannian 2-disks and 2-spheres.

For example, I will prove that on any Riemannian 2-sphere there M exist at least three simple periodic geodesics of length at most 20d, where d is the diameter of M, (joint with A. Nabutovsky, Y. Liokumovich). This is a quantitative version of the well-known Lyusternik and Shnirelman theorem.