Real algebraic curves and abelian varieties: from Comessatti's theory to solutions of the KP equation

The theory of real abelian varieties and real jacobians traces its roots in the old works by Comessatti (1925-1926). We begin by explaining the basic ideas from this beautiful theory that has been rediscovered several times in the literature. We then show how these ideas can be used to reprove and improve classical theorems by Dubrovin-Natanzon on the reality and regularity of quasiperiodic solutions of the Kadomtsev–Petviashvili equation, which can be used to model shallow water waves. Based on joint works with Kummer-Plaumann, and Celik-Fevola-Kummer.