Introduction to Algebraic Geometry

Algebraic geometry is where polynomial equations and geometric shapes become the same object: you study curves and surfaces by studying the rings of functions on them, and vice versa. Expect five problem sets and two take-home midterms that walk you through Nullstellensatz, Bezout, intersection multiplicity, divisors, and culminate in Riemann-Roch and the classical 27 lines on a cubic surface. It sits on top of a solid commutative algebra background and opens the door to modern algebraic geometry (schemes, sheaves, Hartshorne proper), number theory, and complex geometry — basically the gateway course if you want to do research-level pure math.