Lectures:  Exercises:  
Lecture 1, Tuesday, September 15, 2016 Definition of Riemann surfaces. First examples. This lecture was given by Dr. Sara AriasdeReyna . 
Exercises from Lecture 1 psfile , pdffile  
Lecture 2, Tuesday, September 22, 2015 Holomorphic functions, holomorphic maps, meromorphic functions, first properties of morhisms of Riemann surfaces. 
Exercises from Lecture 2 psfile , pdffile  
Lecture 3, Tuesday, September 29, 2015 Meromorphic functions as holomorphic maps to the Riemann sphere, local behaviour of holomorphic maps. 
Exercises from Lecture 3 psfile , pdffile  
Lecture 4, Tuesday, October 6, 2015 Local behaviour of holomorphic maps: first corollaries. Fundamental group, topological classification of compact Riemann surfaces. 
Exercises from Lecture 4 psfile , pdffile  
Lecture 5, Tuesday, October 13, 2015 Topological classification of compact Riemann surfaces. Degree of a holomorphic map between compact Riemann surfaces. Number of poles and zeroes of meromorphic functions on compact Riemann surfaces. 
Exercises from Lecture 5 psfile , pdffile  
Lecture 6, Tuesday, October 20, 2015 Divisors. Sheaves of modules associated to divisors. 
Exercises from Lecture 6 psfile , pdffile  
Lecture 7, Tuesday, October 27, 2015 Divisors and invertible sheaves. RiemannRoch space and its finite dimensionality. 
Exercises from Lecture 7 psfile , pdffile  
Lecture 8, Tuesday, November 3, 2015 Cotangent space, differentials. Holomorphic differential forms. 
Exercises from Lecture 8 psfile , pdffile  
Lecture 9, Tuesday, November 10, 2015 Meromorphic differential forms. Relation between global meromorphic functions and differential forms. Genus of a compact Riemann surface. The RiemannRoch formula. The RiemannHurwitz fomula. 
Exercises from Lecture 9 psfile , pdffile  
Lecture 10, Tuesday, November 17, 2015 Coverings and universal coverings. Holomorphic maps of complex tori. Isomorphism classes of complex tori. 
Exercises from Lecture 10 psfile , pdffile  
Lecture 11, Tuesday, November 24, 2015 Automorphisms of complex tori. Meromorphic functions on complex tori. 
Exercises from Lecture 11 psfile , pdffile  
Lecture 12, Tuesday, December 1, 2015 Complex tori as algebraic curves. jinvariant. 
Exercises from Lecture 12 psfile , pdffile  
Lecture 13, Tuesday, December 8, 2015 Integration of differential forms along curves, residue theorem and its inverse. 
Exercises from Lecture 13 psfile , pdffile  
Lecture 14, Tuesday, December 15, 2015 Jacobian of a compact Riemann surface of positive genus. AbelJacobi theorem and its first corollaries. Compact Riemann surfaces as projective algebraic varieties. 
Exercises from Lecture 14 psfile , pdffile 
Fundamental group. A chapter of the Algebraic topology book by Allen Hatcher.  
Topological classification of surfaces by Richard Koch. 

A note on summability (in German) by Günther Trautmann. 
Forster, Otto.  Lectures on Riemann surfaces; transl. by Bruce Gilligan, New York: SpringerVerlag, 1999.  
Schlichenmaier, Martin.  An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces, Berlin: Springer, 2007.  
Freitag, Eberhard; Busam, Rolf.  Complex analysis. Transl. from the German by Dan Fulea, Berlin: Springer, 2009.  
Freitag, Eberhard.  Complex analysis 2. Riemann surfaces, several complex variables, Abelian functions, higher modular functions, Berlin: Springer, 2011.  
Miranda, Rick.  Algebraic curves and Riemann surfaces, Providence RI: American Mathematical Society, 1995.  
Farkas, Hershel M.; Kra, Irwin.  Riemann surfaces, Springer, 1992.  
Griffiths, Philip; Harris, Joseph.  Principles of algebraic geometry, New York : J. Wiley, 1994. 