I’m currently speaking with a professor about doing an independent study course in complex analysis. I took an undergraduate course with him over the summer which went through the standard material in an introductory course (defining the complex numbers, a little bit of metric topology on the complex plane, differentiation, integration, power series, Laurent series, and residue theory; specifically, Chapters 1 – 7 in Gamelin). I had an idea originally of just going through a graduate-level book on the subject, but the professor seemed more keen on targeting a specific topic and reading through sources in that. I also have to do a presentation during Finals Week relating to the topic. Originally, I had the idea of presenting on either the homotopic and homological properties of complex integration or do it about automorphism groups. The first idea doesn’t seem like it fits well with the idea of focusing on a topic. Does anyone have any ideas for independent study topics?
For a bit of context as to my background, I’ve done courses in group theory and ring theory at the level of Artin, commutative algebra at the level of A&M (chapters 1-3, 5), number theory at the level of Niven/Zuckerman/Montgomery, and single variable real analysis at the level of Baby Rudin. I don’t have any particular interest in my field, but I really enjoy it when fields of math intersect. For example, my commutative algebra class was also an independent study course, and my presentation was on the module of Kähler differentials.