Suggestion for a good book on Riemann Surfaces - your personal experiences

[Mathmos6]

Hello everyone - I'm a third year student at Cambridge university, and I've recently started taking a course on Riemann surfaces along with a number of other pure courses this year.

The problem is, the lecturer of the course is of a rather sub-par standard - whilst I don't doubt he's probably exceptional in his field, he often struggles for clarity, and the work we have to do ourselves is often barely, if it all, linked to the material on the lectures, or indeed quite often tenuously so at best.

The following - http://www.dpmms.cam.ac.uk/study/II/Riemann/2010-2011/RS10.2.pdf - is an example of one of the problem sheets I'm currently working on, and http://www.dpmms.cam.ac.uk/study/II/Riemann/2009-2010/RS09.3.pdf is last year's version of the work I will have to do in a few weeks' time. The small majority of the questions I -can- do are Algebraic Topology which I've taught myself - in general I'm really struggling to find ways to utilise the lecture notes effectively to understand a lot of this, and so I'm looking for a book which might help me get to grips with some of the concepts I might need to answer questions like those in the above PDFs. From your experience, is there anything at all you might be able to recommend which you yourself used, a book which looks like it might be relevant for some or all of the above problems?

I'm well aware there may be no such book, but I thought I would ask anyway in case you had something which you found extremely useful in getting the hang of a lot of the relevant concepts, and which might enlighten me on how one might approach a few questions like those above. We do have a recommended book list, but it is evident the lecturer is unaware of the chasms between his lectures and his worksheets, so I don't have a lot of faith in them, but I'd be very glad to look into anything relevant you PF users have read in the past! As I said, this is with an eye primarily to solving problems like the above - in terms of learning about what a Riemann surface is etc., I'm happy with the lecture notes, it's applying the results to the above problems where I come undone. Any thoughts? Thanks very much for the help, any suggestions welcome! (And indeed multiple such suggestions, assuming no one book will cover all of the above) - Mathmos6

 

[mathwonk]

You may be asking a slightly "bad" question in a pedagogical sense. I.e. you seem more concerned with satisfying some requirements than in learning the material. Those are really excellent problems, and are obviously intended to give you a great learning experience if you struggle with them on your own. They suggest that your instructor is really outstanding at choosing learning materials for you. They also suggest he has high expectations of your effort, as they are not at all trivial.

If you listen to yourself you may also hear the usual students error. I.e. you are claiming to understand very well from the class lectures what a Riemann surface is, and yet you say you cannot do the problems. Well, that means you do not understand the lectures as well as you are expected to do, or rather that you have not augmented your learning by doing the work expected of you. Some of those problems even tell you specifically which theorems to imitate in their solution. It is obvious that they have been carefully chosen to help you learn to use the ideas in the class notes.

Because of this, you may do yourself a disservice to try to find a source that will weaken the experience of thinking about them yourself. You are obviously expected to have a good grasp of covering spaces and basic topology as well as one variable complex analysis. Some of those problems are ones I have assigned in my undergraduate first complex analysis course. If you have that background, rather than seek book resources for those problems, it sounds from their language as if it is permissible to get help or advice from the course instructors. I suggest doing that, i.e. asking for hints or suggestions, not solutions. Another good practice is to work together with one or more other students to help each other generate ideas, assuming this is acceptable by your instructor. It is surely better than copying answers from a book.

Having said that, and I apologize for lecturing you, you may find some help from reading the book by Rick Miranda, Riemann surfaces and algebraic curves. This was the text for my own course in Riemann surfaces last spring, and I found it excellent. It will be most helpful if you use it to further understand the basic ideas rather than searching in it for solutions to your own homework.

From your question, it seems you may not be interested in this, but I will post as well my own first day summary of the whole first course in Riemann surfaces. This will not help you do any specific problems, but may help you (and others) grasp the point of the subject. Since you are under time pressure, you might just glance over it now, and reread it after the course ends, or save it entirely for later.

 

[mathwonk]

To get you started, let me offer you a few more sets of notes from my undergraduate course in complex variables. These solve some problems similar to but not quite as general as some of those in your problem assignments.

Thus I hope these serve as hints or models for your own work. For instance, the notes here on QFT's solve the degree 2 case of one of your problems on mappings defined by rational functions. The notes on the fundamental theorem of algebra are also related to but easier than the general rational mapping problem.

Thus you cannot copy these solutions for your own, but you may be able to get guidance from them. I hope this is legitimately helpful, and does not go too far. It may be appropriate as well to acknowledge on your work any help you get from these notes, if you use them, as well as any help you get from other sources, including books and classmates.

As a teacher, your instructor is also interested in knowing just how feasible it is for his students to work his homework with the guidance he has provided. And he wants to know what is going through their minds as they reflect on his presentation. Good luck.

 

[mathwonk]

have you searched the resources available on your own departmental web site?

surely these are admissible, but you might ask, since these do work some of your problems.

e.g.

http://www.dpmms.cam.ac.uk/~agk22/RS-notes.pdf

see theorems 2.15, 19.56.

 

[mathwonk]

By the way, may I ask who's teaching your course, Kovalev? Shepherd-Barron?

You should chat with the instructor, and/or one of those guys. They can be very helpful.

 

[Mathmos6]

You may be asking a slightly "bad" question in a pedagogical sense. I.e. you seem more concerned with satisfying some requirements than in learning the material. Those are really excellent problems, and are obviously intended to give you a great learning experience if you struggle with them on your own. They suggest that your instructor is really outstanding at choosing learning materials for you. They also suggest he has high expectations of your effort, as they are not at all trivial.

If you listen to yourself you may also hear the usual students error. I.e. you are claiming to understand very well from the class lectures what a Riemann surface is, and yet you say you cannot do the problems. Well, that means you do not understand the lectures as well as you are expected to do, or rather that you have not augmented your learning by doing the work expected of you. Some of those problems even tell you specifically which theorems to imitate in their solution. It is obvious that they have been carefully chosen to help you learn to use the ideas in the class notes.

Because of this, you may do yourself a disservice to try to find a source that will weaken the experience of thinking about them yourself. You are obviously expected to have a good grasp of covering spaces and basic topology as well as one variable complex analysis. Some of those problems are ones I have assigned in my undergraduate first complex analysis course. If you have that background, rather than seek book resources for those problems, it sounds from their language as if it is permissible to get help or advice from the course instructors. I suggest doing that, i.e. asking for hints or suggestions, not solutions. Another good practice is to work together with one or more other students to help each other generate ideas, assuming this is acceptable by your instructor. It is surely better than copying answers from a book.

Having said that, and I apologize for lecturing you, you may find some help from reading the book by Rick Miranda, Riemann surfaces and algebraic curves. This was the text for my own course in Riemann surfaces last spring, and I found it excellent. It will be most helpful if you use it to further understand the basic ideas rather than searching in it for solutions to your own homework.

From your question, it seems you may not be interested in this, but I will post as well my own first day summary of the whole first course in Riemann surfaces. This will not help you do any specific problems, but may help you (and others) grasp the point of the subject. Since you are under time pressure, you might just glance over it now, and reread it after the course ends, or save it entirely for later.

I apologise if it sounded like I was trying to avoid doing the work or learning from the problems myself but instead trying to find a book from which to copy them - indeed, precisely the reason why I didn't ask any specific questions on the problem sheets is because I want to do them myself in order to learn as much as I possibly can from them.

The issue I am having is that the lecturer (P M H Wilson, author of the geometry text 'Curved Spaces', since you asked) is giving very sparse examples, and whilst I can grasp the concept of -what- a Riemann surface is at the basic level (one of the few situations an example was provided for), I have trouble actually applying the results to some of the problems - indeed, we are just proving a series of theorems and not really seeing any applications in this sense. I wasn't hoping to find direct solutions to these questions, just a few similar situations in which, for example, the monodromy theory could be applied to a problem.

Whilst I do have resources such as supervisors and lecturers to approach if needs be, I am very much of the mindset that I will appreciate the method of solution to a problem more if I am able to derive it myself, as you said. All I was hoping for really was a source of a few somewhat-similar examples to get me on my way, and once I have the hang of the easier questions I can hopefully approach the harder ones myself with the knowledge I've built up. In addition, the course is very comparatively under-subscribed, perhaps due to the lecturing style or perhaps because the room in which lectures are given is very badly laid out and makes the lectures quite an unpleasant experience in comparison to a number of the other lecture rooms - as such, the only friends I have who are also attending the course are struggling with it far more than me! I have worked with them previously but it tends to be the case that I simply end up explaining the problems I completed fully and not really getting much back.

Anyway, I didn't mean to come across as seeking the easy way out - I'm actually taking more courses than anyone else I've spoken to in our year because I'm enjoying them all so much, though it does make me a lot more pressed for time on some of the worksheets. I was simply hoping for assistance in making the first step between 'here is a list of theorems' and 'here are some results you can prove somehow via your list of theorems'. I will have a look at the Miranda text and see if it is of use - I've been teaching myself algebraic topology from Hatcher so those questions don't tend to be a problem, but I'll see if I can make some use of the resources you suggested - thanks for replying!

(Oh, and I am certainly interested in Riemann surfaces, though I was originally hoping the course would take a more visual approach to the topic - however, judging from the schedules I think that was optimistic of me. At any rate, I enjoy complex analysis, so I will persevere.)