Abstract

To paraphrase Jacques Hadamard: What a partial differential equation is, is well known. What hyperbolicity is, will be explained. I will give an expository discussion of systems of partial differential equations and their local well-posedness, with focus on the notion of regular hyperbolicity due to Demetrios Christodoulou; in particular the role of energy estimates will be emphasized and their systematic construction will be described. Within this framework, a local causal structure of hyperbolic equations can be defined, extending the concept of hyperbolic development of Jean Leray. Finally, I will touch on some recent work in which this local causal structure is compared against the background space-time causal structure in the context of Lagrangian theories of maps on a Lorentzian manifold.

The talk will be accessible to non-experts, and Part III and CCA students are encouraged to attend.