Math 416/816: Algebraic Curves and Riemann Surfaces

Lectures:   Are in Slot 14 (Tuesday 11:30-12:20, Wednesday 1:30-2:20, and Friday 12:30-1:20), in Room 110.

Office Hours:   The hour following class (Tuesday 12:30--1:20, Wednesday 2:30-3:20, and Friday 1:30-2:20) or by appointment.

About the course

Riemann Surfaces were introduced in the 19th century to help resolve strange formulae and explain mysterious behaviour arising from problems in function theory. Riemann's initial discription was confusing to many people and the attempt to understand these objects and his wonderful new ideas led to the growth or creation of many fields of mathematics: topology, the theory of manifolds, potential theory, and algebra to name a few.

The idea of a Riemann surface is now a central one in mathematics, and appears in such seemingly diverse areas as integrable systems, number theory, algebraic geometry, and string theory.

Because of the many points of view on Riemann surfaces, and the large number of other parts of mathematics that they interact with, there are a lot of different approaches, and topics we could cover.

My goal is to touch on several of these areas and show how they relate. In particular, I would like to concentrate on the interaction between the complex analytic, classical geometric, and commutative algebraic points of view.

The organization of the course will largely follow the natural flow of ideas between these subjects.

We will first try to understand Riemann surfaces as complex analytic manifolds. We want to understand the basic topological properties of these surfaces, and of the maps between them.

We will then look at curves as submanifolds of complex manifolds, in particular as submanifolds of the projective plane. This will lead to the beautiful classical theorems in the plane: Bezout's theorem, and the adjunction formula.

The description as subvarieties of the projective plane will allow us to treat the curves purely algebraically. We'll study how to describe the curves using only algebraic objects like rings and ideals, and how to express maps between curves in the language of rings.

Finally, we'll largely return to the manifold point of view and understand the final fundamental theorem about algebraic curves: the Riemann-Roch theorem, which governs the geometry of curves in subtle ways. We will also look at applications of this theorem to geometric questions.

Grading Scheme

Anything we have to learn, we learn by the actual doing of it.

-- Aristotle, Nicomachean Ethics, Book II.

As with many things in mathematics, most of the ideas involved in studying Riemann surfaces are not that complicated, and our difficulties with them are mostly psychological: we we have to go through an awkward period of confusion before realizing how simple and straightforward the ideas were ''all along''.

The usual cure for the confusion brought on by abstract ideas and new definitions is to do a few concrete computations. In order to encourage everyone to do computations, I'm putting most of the grading weight of the course on the homework. The grading scheme is

Homework 75%
Final 25%

The homework questions will be designed to give you a chance to play with some of the concepts we've been talking about in class. For those who want more practice problems, verifying uncompleted or omitted calculations from the lectures is a good place to start.

Homework Assignments

Assignment 1: Practice with coordinate charts:   .ps   .pdf  .dvi
Assignment 2: Branched covers and the Riemann-Hurwitz formula:   .ps   .pdf  .dvi
Assignment 3: Homogeneous and nonhomogeneous polynomials:   .ps   .pdf  .dvi
Assignment 4: Verifying details of the proof of Bezout's theorem:   .ps   .pdf  .dvi

Final Exam: The exam is due Wednesday, December 17, 2003 .ps  

Lecture Notes

I was going to attempt to put sketchy TeX versions of the lecture notes here, but I now realize that I will never manage to get this done. Instead I'm just going to use it to outline the lecture topics.

The plan for future lectures is flexible -- I'll be moving it around a bit depending on how things are going in class.