Algebraic geometry studies geometric shapes using algebraic methods. But somehow differential equations show up! I will give an overview of what algebraic geometers (at large) care about while highlighting the role analysis plays: for example, hodge theory uses elliptic equations to study topology of complex manifolds, and Monge–Ampère equations are fundamental in the study of Calabi—Yaus, the so-called hidden dimension of the universe.

Conversely, AG also gives back to differential geometry, for example by giving algebraic criteria for existence of canonical metrics and helping with moduli problems. This talk is intended to be an introduction to algebraic geometry with a PDE twist, and no background in either field is needed.