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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-Fries 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.

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.

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-Fries 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.

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-Fries 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.

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-Fries 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.

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.

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-Fries 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.