Time and Place: University of Trento, Summer Term 2027. Precise time and date to be announced.
Audience: Doctoral Students in Mathematics.
Abstract: "Hyperbolic partial differential equations (PDEs) play a fundamental role in mathematical physics, as they model many physical phenomena, particularly those involving wave propagation, causality, and finite propagation speed of information. This course is devoted to the analysis of such equations in the global setting, i.e.~on (globally hyperbolic) Lorentzian manifolds. We begin with the necessary background, introducing basic concepts of Lorentzian geometry, which provides the natural geometric framework for the study of hyperbolic PDEs, and the analysis of differential operators on manifolds. We then study two important classes of hyperbolic PDEs in detail: symmetric hyperbolic systems, a large class of first-order operators, and normally hyperbolic, i.e.~wave-type, operators. Particular emphasis will be placed on well-posedness of the Cauchy problem and on the construction and properties of advanced and retarded Green operators, which encode the causal propagation of solutions. Throughout the course, we illustrate the general theory with examples arising in relativistic field theory, including the Klein-Gordon and (spin) Dirac operators. Only a basic knowledge of differential geometry and (functional) analysis is assumed."
Lecture Notes: to appear.