This will be a two-talk mini-series. In the first talk I will give a very gentle introduction to Brill–Noether theory for complex algebraic curves. I will start from
the basic question: what kinds of functions can exist on a curve, and how many independent ones can we hope for?

After introducing some necessary concepts, I will explain the classical results of Brill–Noether theory, which give a simple criterion for when such functions should exist on a “typical” curve. The emphasis will be on ideas and examples rather than technical details.