I am a master's student planning to write a master's thesis on Riemann surfaces. I plan to study Forster's Lectures on Riemann surfaces. What side topics could one study to spice up the thesis? I am particularly interested in analytic aspects.

Added later.. Could anyone suggest how much background would one require to understand geometry of algebraic curves by authors such as Harris, Arbarello, etc... I know already Complex algebraic curves by Kirwan, except for the last chapter on singular curves. Which books would provide me enough materials so that I can start feeling comfortable with the book

Geometry of Algebraic Curves by authors Harris, Arbarello,Griffiths,Cornalbla

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    $\begingroup$ What's so bland about Riemann Surfaces that requires spicing up? $\endgroup$ – stankewicz Apr 24 '14 at 11:26
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    $\begingroup$ I only had a quick glance at the TOC of Foster, but did you compare Gunning's book? $\endgroup$ – Tom Bachmann Apr 24 '14 at 13:54
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    $\begingroup$ Moduli stuff would be good. Maybe Teichmueller spaces too. $\endgroup$ – user40276 Apr 24 '14 at 16:15
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    $\begingroup$ Well Griffiths and Harris does all those topics in your comment before Riemann surfaces, although this may be a bit of a hard road to take. Also look at Donaldson's new book on Riemann surfaces. I taught a course from it last year, and it was a lot of fun, at least for me. $\endgroup$ – Donu Arapura Apr 24 '14 at 17:13
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    $\begingroup$ I guess it is a bit bland if you omit theta functions, the theta divisor, Riemann's singularity theorem, Torelli's theorem, the theory of Andreotti and Mayer, Brill - Noether theory, and Mark Green's theorem,. You might consult Geometry of algebraic curves, by Arbarello, Cornalba, Griffiths, and Harris, for most of this. My favorite proof of the RST, due to Mumford and Kempf, is exposed in the appendix of preprint #9 on this page: math.uga.edu/~roy $\endgroup$ – roy smith Apr 26 '14 at 1:43
  1. Forster just touches the Riemann-Hilbert problem and fiber bundles. Expansion on this can be interesting I recommend the books of Bolibrukh.

  2. Applications of compact Riemann surfaces to solitons ("Explicit solutions" of the Koreweg-de-Vries equation etc. In a comprehensive course of algebraic curves and Riemann surfaces taught by Drinfeld, that I took in early 1980-s this was included as an example of application. (Exposition was based on Krichever's papers which were new at that time. Now you can find this in many books).

  3. Belyi theorem was used in this course as a HW exercise, but since then much interesting stuff was added to this.

  4. Myself, I use holomorphic dynamics to "spice" my Riemann surface courses, also Kleinian groups. Especially Sullivan's proofs of the finiteness and non-wandering theorems.

  5. Theta-functions and the explicit solution of the inversion problem are not mentioned in Forster, though this material is due to Riemann himself. This in turn has a lot of applications, in particular, to item 2 above. On this I recommend Mumford's classical Tata lectures on Theta.

EDIT. Some References

  1. MR1276272 Anosov, D. V.; Bolibruch, A. A. The Riemann-Hilbert problem. Aspects of Mathematics, E22. Friedr. Vieweg & Sohn, Braunschweig, 1994. (There are many books of Bolibrukh in Russian, also his survey papers in Rus. Math. Surveys etc).

  2. The key author is Dubrovin, with various co-authors. See the reference to his lectures in the comments below, and also his papers in Russian Math. Surveys., and books.

  3. MR1305390 The Grothendieck theory of dessins d'enfants. Papers from the Conference on Dessins d'Enfant held in Luminy, April 19–24, 1993. Edited by Leila Schneps. London Mathematical Society Lecture Note Series, 200. Cambridge University Press, Cambridge, 1994.

  4. MR2193309 Milnor, John Dynamics in one complex variable. Third edition. Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006. Also: MR0819553 Sullivan, Dennis Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains. Ann. of Math. (2) 122 (1985), no. 3, 401–418.

  5. Mumford, Tata lectures on Theta.

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    $\begingroup$ +1 for soliton solutions to integrable PDEs, which I was about to mention as well. Boris Dubrovin's notes on the subject (people.sissa.it/~dubrovin/rsnleq_web.pdf) include a self-contained introduction to Riemann surfaces, suggesting that the material is elementary enough to talk about in a thesis which is ultimately about Riemann surfaces. @AlexandreEremenko, can you name some of the books you mentioned? I've been trying to learn about this stuff for a while, and I'm having a hard time finding references. $\endgroup$ – Vectornaut Apr 24 '14 at 16:15
  • $\begingroup$ thanks for this reference to "Solition".i was thinking of this $\endgroup$ – Koushik Apr 25 '14 at 10:03

You could work out a syntactical axiomatization such that each of

  • Riemann surfaces and holomorphic maps

  • (Nonsingular) algebraic curves and morphisms

  • Quasiconformal surfaces and quasiconformal maps

is a model of your axioms.

  • $\begingroup$ what do you mean by "syntactical axiomatization" could you elaborate? $\endgroup$ – Koushik Jul 13 '14 at 15:35

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