I will explain the concept of Finsler metrics as a generalization of Riemannian metrics. The word 'projective' in this context means that we forget about the parametrization of the geodesics and consider them just as oriented pathes. For example, two metrics are called projectively equivalent, if they have the same geodesics up to orientation preserving reparametrization. A vector field that takes the geodesics of a Finsler metric to geodesics as pathes is called a projective symmetry.
Depending on time, I will discuss three projective problems on Finsler surfaces:
1. Describe Finsler metrics with 'many' projective symmetries.
2. Given a system of pathes, find all Finsler metrics with this geodesics.
3. Topological obstructions for the existence of projective equivalent metrics on closed surfaces.