Hyperbolic systems and propagation on causal manifolds

TL;DR

This survey explores solving the global Cauchy problem for hyperbolic PDE systems on causal manifolds using microlocal sheaf theory, extending classical results with algebraic and geometric tools.

Contribution

It introduces a microlocal sheaf-theoretic approach to hyperbolic systems on causal manifolds, broadening the scope beyond classical methods.

Findings

Successful formulation of the global Cauchy problem in this framework

Extension of classical theorems using microlocal sheaf tools

Framework applicable to a wide class of hyperbolic PDEs

Abstract

This is essentially a survey paper in which we solve the global Cauchy problem on causal manifolds for hyperbolic systems of linear partial differential equations in the framework of hyperfunctions. Besides the classical Cauchy-Kowalevsky theorem, our proofs only use tools from the microlocal theory of sheaves, that is, tools of purely algebraic and geometric nature.