This is an informal and friendly tutorial about algebraic geometry, designed for analysts. Little background about algebra is assumed. We will mostly use examples to illustrate ideas. This talk will be accessible to second year graduate students.

Given a system of polynomial equations in several unknowns, it is possible to find all the common zeros. In contrast, nobody has been able to find all the zeros of the Riemann zeta function. In this talk, I will explain why it is possible to solve a system of polynomial equations in several variables, and the structure of the set of common zeros.

Hironaka, the first person who introduced Blow-ups in 1964, actually got a Fields Medal in 1970. It took him TWO ENTIRE issues in the Annals of Math to do a rigourous job, we will settle for a fun example instead.

This talk should at least be mildly entertaining, if not educational.