To be added: 

Fall 2024 - Riemann Surfaces

Spring 2025 - Commutative Algebra

Directed Reading Course Abstract.

We shall be reading "Ideals, Varieties, and Algorithms" by Cox, Little, and O'Shea. The general plan is to cover most of the material in the earlier chapters. A link to an online pdf can be found here. The goal would be to cover Chapters 1 and 2 which is roughly 100 pages.

I think the best way to do a DRP like this is to have someone give a talk each meeting.
Prerequisites: Mathematical maturity. I expect that most people will need a minimal level of background in algebra. The book is quite readable.

Meeting Times.

Fridays in Room 903 at Van Vleck Hall. Each student is expected to give talks over the course of the semester.

Directed Reading Course Abstract.

We shall be reading "Algebraic Curves and Riemann Surfaces" by Rick Miranda. The ambitious goal will be to cover most of Chapters I-VI which has the proof of the Riemann-Roch Theorem and a statement of Serre Duality. The course itself should serve as a gentle introduction to many notions in basic algebraic geometry as well as the deep connections between complex geometry and algebraic geometry. The first chapter covers the basic definitions of Riemann surfaces (as second countable Hausdorff spaces with a complex structure) and standard examples. The following three chapter present examples of Riemann surfaces, the study of functions on these surfaces, and operations that can be done on them such as integration. For integration, we shall discuss integration on surfaces (which is helpful intuition if one wants to study integration on manifolds) and we prove a version of Stoke's Theorem, the Residue Theorem, and utilize homotopy and homology. Afterwards, Chapters V and VI cover the theory of divisors and meromorphic functions which can be used to specify maps or embeddings into projective space and classify these surfaces based on a notion of "genus" (in fact, there are three proven equivalent definitions).

The course itself will barely scratch at the connection between Riemann surfaces and algebraic curves, but it should equip students with enough background to pursue that connection on their own time.

The student should expect approximately 2-4 hours of reading each week on their own time. During our meetings, I shall ask questions in an attempt to provoke student insights and ask that students come to the DRP meetings prepared with some questions.