Geodesic Lévy flights on Zoll surfaces

In this talk I will present recent results concerning the expected time for a geodesic Lévy flight on a Zoll surface to reach a ball shrinking to a point. As we shall see, the leading-order asymptotics are universal, but the first correction term encodes subtle geometric information. In particular, I will show that this correction is determined by the singularity type of the conjugate locus, measured by the cuspidal degree of a conjugate point. This is joint work with E. J. Godfried and L. Tzou.