MAST90056: Riemann surfaces and complex analysis

3 assignments, 20% each and exam 40%

This subject is the meeting point of topology and complex analysis. We learn a little bit more complex analysis but mainly study these objects called Riemann surfaces which are surfaces with some additional complex structure, loosely speaking.

This will be a review to help you decide whether or not to take this subject.

To give a useful review, lemme start with the pros. Riemann surfaces are interesting objects to study in its own right and Paul is a strong mathematician, quite knowledgeable who is fair in his marking and believes in conscientiousness. Provided that you honestly worked through the assignment as best you can, he'll give you a good mark that you will deserve, regardless of anything else (I'll leave you to interpret that however you like). My understanding is that Paul likely knows that this subject is a little too difficult (in the wrong way) and adjusts his marking accordingly, which is always nice to see. The exam was a beautifully written exam for a 3 hour session but since we were only given 2 hours to do it, I found it to be extremely difficult, but I believe was scaled which was very fortunate. He is also very willing to help students, always staying back after lectures for as long as needed to answer questions and giving 3 hours of consultations per week which is quite a lot I reckon.

With that said, to save you some time let me just say that if you're just looking for some random maths subjects to do for your Msc then do not pick this subject. Otherwise, if you know some differential geometry and either have a specific reason to study Riemann surfaces or are interested in it then this subject is for you. There are a lot of problems with this subject, particularly its structure, the way it is taught and the assignments. Before I go into this, lemme just say that yes I did ok in this subject but not without a lot physical and mental pain throughout the semester. I had to pay a high price for it, let's put it that way.

As for the structure of the course, students are only required to know complex analysis to be able to take this course. However, from the way it is delivered, it is obvious that you do need to know some differential geometry, geometry and topology for sure. I would say that the real prerequisites for this subject are metrics and hilbert spaces, geometry, complex analysis and differential geometry. My understanding from what the lecturer said is that if he puts all that in the description then the number of students taking this subject will be too low. He said that we are expecting a different kind of maturity here, meaning that students should take the subject and drop it if they find it not to be their cup of tea at around week 3. There are obvious flaws in this argument imo but I didn't wanna start a debate with him. In any case, if you take this subject you can imagine its like taking on a subject with twice as much workload since you have to be teaching yourself topology and differential geometry at the same time. The students who have had all the background knowledge (most definitely not me) will find this subject to actually be pretty chill.

As for how it is taught, like I said Paul is a strong mathematician but having been at the top for so long he likely forgot what it was like to be a student and is imo not very organised and horrible at explaining/motivating concepts. There are no prewritten notes and he pretty much handwrites everything as we go which I do not mind too much. My problem with this is that the statements that Paul writes down are very terse and in many important cases not rigorous and even wrong which really makes the studying process a lot more daunting. At times I would wonder if I don't understand the material or the statement was just wrong. In addition, each lecture will consists of him just pumping out content for us to learn, without any motivation or at best just some obvious explanation that certainly is insufficient for a student. In many cases he'll just give some terse explanations so that you can go through the book yourself to learn the rest of the details. This again is understandable and one would expect that a Msc student can do such a thing but it certainly takes a lot of time for those with weak backgrounds and together with all the other factors piling on top of one another, it is not helpful for students' learning imo.

Finally for the assignment, I understand that they are worth 60% of the course and so they are meant to be more difficult. My opinion though is that they are too difficult and we are given like a week to do them which is far too short. Paul said it himself that the assignments are designed to hurt you and make you better for it. I feel that the difficulty level was a little too great for us, especially considering how most of us, like myself, did not have much background in differential geometry (let alone those without a background in topology). In addition, Paul said it himself that despite giving us a week to do it, he does not expect people to spend a week working on it, meaning that his intention is to have us work on it for like 3 - 4 days and that's it. I, however, had to spend like an entire week only on that assignment, could not study any of my other subjects which puts me really far behind. Idk maybe I'm just dumb but again my opinion is that the assignment was far too difficult in many many ways. Btw, one of the assignment required you to read one of the reference book and explain it so that's another reason to grab the books.

The bottom line is, unless you're well prepared in advanced with all the background knowledge and studying skills (you'll be left to your own device for most of the semester due to the unreliability I've stated above), you'll be playing an unfair game against you, where even basic things like precision, correctness and rigour of statements will be scarce lol

Science

Yes, with screen capture

Paul Norbury

Yes, 2 but useless, I did not use them and will never recommend them

1 Out of 5

Ahlfors and Forsters book. Definitely recommend for reasons I will discuss. But it's up to you whether to get it or not, of course.

On the surface, 3 one hour lectures

2021 Semester 2

92