Title: Basics on simply-connected closed smooth 4-manifolds and compact algebraic surfaces.     Abstract: The classification of compact algebraic surfaces, which goes by the name of Enriques–Kodaira-Mumford-Bombieri classification, is a classical topic in AG.   Underlying each complex surface is a smooth 4-manifold which could have infinitely many pairwise distinct smooth structures. In practice, what we often do is to first pin down the exact homeomorphism type via application of Freedman’s Theorem.   In this talk, I will explain the Geography lattice and the problem of botany and how to go about figuring out the homeomorphism type via intersection forms.   P.S: By the remarkable work of Donaldson, every closed symplectic 4-manifold admits a structure of Lefschetz pencil, which can be blown up at its base points to yield a Lefschetz fibration. Understanding symplectic 4-manifolds is equivalent to understanding genus g Lefschetz fibrations over CP^1.