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I have been told by different people that probably no subject can unify algebra, analysis and geometry better than Riemann surfaces. Regardless of how true it is, I'm looking for a textbook that explains the basic ideas and theorems of Riemann surfaces with a fairly reasonable background which includes undergraduate algebra, undergraduate analysis and undergraduate geometry.

In other words, the audience of the book should be advanced undergrad students. Since I want it for self-study, I'd really like to find a book that has solutions. If not, then a textbook with graphics, drawings or intuitive explanations would suit me the best.

I'm tagging this question as 'reference-request' and 'soft-question'. I will really appreciate it if you share with me your pedagogical experience or your own troubles when you wanted to get introduced to Riemann surfaces. Any piece of advice about how to approach the subject is welcome and highly appreciated

stressed out
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1 Answers1

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Take a look at the last chapter of Gamelin "Complex Analysis". The book is aimed at UCLA undergraduate students and it has exercises.

One more option is:

Narasimhan, Nievergelt, "Complex Analysis in One Variable". It is aimed a bit higher than Carleson and Gamelin (and is aimed at 1st year graduate students), but it covers more material.

user682141
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Moishe Kohan
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  • Thanks (+1). I checked the book. Could you please tell me which chapters are necessary to understand the last chapter? Do I have to read the book thoroughly or can I just ignore some chapters and still understand the last chapter? – stressed out Mar 07 '19 at 04:22
  • @stressedout How much complex analysis do you already know ? – Moishe Kohan Mar 07 '19 at 04:39
  • I have Complex Functions this semester but I have already self-studied the material up to Cauchy's integral theorem and Rouche's theorem. I know the Residue theorem and Laurent series, for example. – stressed out Mar 07 '19 at 04:43
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    @stressedout: Then read chapters 5, 9, 10, 11, 15 and then 16 (Riemann surfaces) of Carleson-Gamelin. You consult other chapters when needed. One thing to know: their book covers only one of the two cornerstones of Riemann surfaces but not the other: They do not discuss the Riemann-Roch theorem. For that I do not know of any undergraduate-level treatment. – Moishe Kohan Mar 07 '19 at 16:22
  • Thank you. If I manage to understand these chapters and finish chapter 16, I will probably read Otto Foster's Lectures on Riemann surfaces or Jurgen Jost's Compact Riemann Surfaces: An Introduction to Contemporary Mathematics. I have heard good things about the later one. Do you think if I understand chapter 16 of T Gomelin's book I will be ready to read and understand those two books I mentioned? Or if you don't know those two books, what do you think my next step should be? – stressed out Mar 08 '19 at 06:34
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    @stressedout: Yes, these are good options but it depends on how much analysis you can handle. Miranda's book "Algebraic curves and Riemann surfaces" is another good choice: Less analysis, more geometry. Yet another option is Farkas and Kra "Riemann surfaces". – Moishe Kohan Mar 08 '19 at 16:11