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In the local existence theory for quasilinear hyperbolic PDE, one key step is to apply Sobolev inequality to obtain uniform pointwise bounds ($L^\infty$) on solutions using $L^2$-based norms on standard Cartesian derivatives. The vector field method, introduced by Klainerman (1985), uses instead a larger class of variable-coefficient derivatives that are adapted to the Minkowski spacetime geometry of the wave equation. This allows one to obtain pointwise bounds that decay-in-time with rates adapted to the light cones in Minkowski spacetime, which can then be used to establish long-time or global existence of solutions with small initial data.

This introductory talk should serve to illustrate the geometric point-of-view used in hyperbolic equations.