Synopsis: The second course in a two semester sequence. The general plan is to cover a couple more topics for classical one variable complex function theory and then spend the rest of the course on Riemann surfaces.
Instructor: Jeffrey Diller (click for contact info, list of my papers, etc.)
Official Time and place: Tuesdays and Thursdays 2-3:15 PM in the Math Bunker (seminar room in basement of Hayes-Healy).
Office hours: After class Tuesday and Thursday, and by appointment.
Textbook: I plan to (mostly) follow a preliminary version of a textbook by my colleagues Mei-Chi Shaw and Chuck Stanton. You’ll find the book in the Google Drive folder I will give you access to at the beginning of the semester. The authors have been kind enough to share their book with us for free and will be very grateful for any feedback you have to offer. So if you spot typos or have suggestions for other improvements, please send them to me so I can relay them to Mei-Chi & Chuck.
There are many books on Riemann surfaces. A couple of other good sources are:
Lectures on Riemann Surface by Otto Forster. I’ve used this as a text in the past.
Curt McMullen’s course notes. A just-the-facts summary sans detailed proofs for the most part. You can find these on his (Harvard) webpage, but I’ll put a copy in the Drive folder for convenience.
Topics list: I hope to cover
Runge’s Approximation Theorem (chapter 5)
The inhomogeneous Cauchy-Riemann equation and its application to proving the interpolation theorems of Mittag-Leffler and Weierstrass (chapter 6)
Basic Riemann-Surface yoga (chapter 7)
Uniformization and applications (chapter 9)
L^2 techniques and harmonic functions (chapter 10)
Compact Riemann surfaces (chapter 11)
Homework: Homework problems will account for 50% of your grade in this course. I'll assign new problems by noon (most) every week or so and give you a week to work on them. Solutions will be due by 11:59 PM Fridays. I don’t plan to write up solutions this semester, but instead I might ask you to revise problems where you made significant mistakes. I strongly encourage you to collaborate with your fellow students when solving homework problems, but you must write up solutions yourself. That is, you should not copy from someone else’s solutions. Tex’d homeworks will be very welcome!
Presentation: each student in the class will give a 30-45 minute presentation, covering some topic related to the class and worth 15% of your grade. You can present something I would’ve talked about anyhow, or something a little outside the topics I’d planned to cover. I’ll count this for 20% of your grade. Ideally a presentation should include some precise definitions and proofs but also some survey a broader range of ideas. I’ll put a partial list of ideas for topics in the Drive folder, and I’d like you to check in with me in the first couple of weeks of class to discuss which topic topic you’ll pursue.
Exams: There will be a single take home final exam for this course, worth 30% of your grade. Ideally I’ll get this up around Easter so you have *lots* of time to work on it. Basically it’ll be a long homework set with problems covering anything from either the first or second semester of the course.
Necessary Background: The first semester of this course ought to suffice.
Any instance of cheating will be dealt with according to Notre Dame’s Academic Code of Honor.
Covid: I don’t intend to have any particular masking policy for class this semester, but please bring any health concerns to my attention. If you believe you have contracted covid (or any other contagious illness that your instructor and classmates would do well to avoid), please let me know and absent yourself until it’s prudent to return. I’ll make sure you find out what happened in the classes you miss.