The marked length spectrum of a closed Riemannian manifold of negative curvature is a function on the free homotopy classes of closed curves which assigns to each class the length of its unique geodesic representative. It is known in certain cases that the marked length spectrum determines the metric up to isometry, and this is conjectured to be true in general. In this talk, we explore to what extent the marked length spectrum on a sufficiently large finite set approximately determines the metric.

I will review the highlights of Strominger’s program of celestial holography, focusing on emerging connections to twisted holography, topological strings and twistor theory. A running example of the talk will be holography for certain self-dual theories placed on an asymptotically flat, scalar-flat Kahler geometry known as Burns space. This is based on work done in collaboration with Kevin Costello and Natalie M. Paquette.