Multiplicity of Closed Geodesics on Finsler Manifolds

The Katok–Anosov conjecture predicts a sharp lower bound for the number of prime closed geodesics on Finsler spheres. The conjecture is known for the two-sphere but remains open in general in higher dimensions. In this talk, I will briefly review progress on this conjecture and related multiplicity problems on compact simply connected manifolds. I will present results for bumpy metrics under additional assumptions, as well as results under curvature pinching conditions. I will then discuss the main tools and explain the main idea of the proof of one of our results. This talk is based on joint work with Huagui Duan.