Around the turn of the 20th century, Castelnuovo and Enriques undertook the classification of complex algebraic surfaces. Their work was later reformulated and completed in the modern language of algebraic geometry by Zariski, Kodaira, Shafarevich, and others. The course will present parts of this classification, with an emphasis on understanding the geometry and topology of algebraic surfaces.

The course is an introduction to the representation theory of infinite-dimensional Lie algebras, focusing on the Lie algebra of infinite matrices. The aim is to understand the Kadomtsev–Petviashvili equation using representation-theoretic methods. This equation exemplifies Integrable systems, a class of differential equations with rich symmetries enabling exact solutions.