Theoretical computer scientists make a regular use of algebraic objects, most prominently polynomials over finite fields, to construct and analyze desired objects such as error correcting codes, expander graphs, pseudorandom generators, cryptographic protocols, and randomness extractors. Reed-Solomon codes is one such classic example. In 1975, Goppa suggested to construct and analyze codes using machinery from algebraic geometry or, more precisely, based on algebraic curves. In a long line of research, such codes were eventually constructed. Surprisingly, these codes even beat the Gilbert-Varshamov bound for sufficiently large fields.

The underlying mathematics being employed for the construction of Goppa codes is far deeper than the standard tools generally used by theoretical computer scientists, and the hope is that it can be further exploited in resolving other open problems, especially problems for which using polynomials one currently obtains the state-of-the-art results. Many important problems fall into this category. For that to happen, the underlying mathematics must be grasped by computer scientists. This is the goal of this course.

In this introductory course we will thoroughly develop the fascinating mathematics underlying algebraic curves to the point where Goppa's framework can be defined and well-understood. The approach we take is mostly algebraic, however, we will not abandon the geometric aspect - we will "think geometrically" (and topologically) and prove theorems algebraically. Students who complete the course should be able to use the mathematical tools developed beyond the code-specific application.

Tentatively, the course will be divided into five chapters:

1. Algebraic curves: curves, rings of functions, resultant, Zariski topology, maximal ideals and points (Hilbert Nullstellensatz), morphisms, singularity.

2. The algebraic aspect - integral closure, multiplication of ideals, Noetherian domains, curves as rings of dimension one, Dedekind domains.

3. Algebraic curves cont.: localization and local rings, the connection between integral closure and singularity, Hilbert's basis theorem.

4. Projective geometry - Discrete valuation rings, the projective plane and projective plane curves.

5. Applications to coding theory - divisors, genus, Riemann's theorem, and Goppa codes.

In the following semester, a follow-up course will be given in which we will continue to develop central tools in the study of algebraic curves such as the Riemann-Roch theorem, the factorization of ideals in ring extensions, ramification and discriminants, the zeta function attached to an algebraic curve, and explicit constructions of algebraic curves with desired properties as well as algebraic-geometric codes. Some of the mateiral may shift from this more advanced course to the introductory course and vice versa.