Arbeitsgruppe Geometrie

Complex Analysis II: Riemann Surfaces (Winter 2013)

This is a course of the Berlin Mathematical School held in English.

News

• [10.10.] In the week from November 25 to November 29 will be held neither a lecture nor a tutorial.
• [23.09.] The lecture starts on October 14, the tutorials start on October 23.

Contents

Riemann surfaces appear in complex analysis as the natural domains of holomorphic functions. Their theory provides powerful tools, examples, and inspiration for such diverse areas of pure and applied mathematics as number theory, algebraic geometry, topology, differential geometry, mathematical physics, and geometric analysis. Riemann surfaces appear in many different guises: as the result of analytic continuation, as algebraic curves, as quotients of a complex domains under discontinuous group actions, as smooth or polyhedral surfaces. This lecture course provides a first introduction to the theory of Riemann surfaces. Motivated by concrete examples and applications, the following topics will be treated: topology of Riemann surfaces, holomorphic maps, coverings and branched coverings, meromorphic functions on a Riemann surface, elliptic curves and elliptic functions, abelian differentials, theorems of Abel and Riemann/Roch, theta functions, discrete Riemann surfaces.

Homework policy

To get a certificate for the tutorial, you need to satisfactorily complete 50% of the homework assignments. Homeworks shall be prepared in groups of two or three students.

Exercise sheets

• Exercise Sheet 1
• Exercise Sheet 2
• Exercise Sheet 3
• Exercise Sheet 4
• Exercise Sheet 5
• Exercise Sheet 6
• Exercise Sheet 7
• Exercise Sheet 8
• Exercise Sheet 9
• Exercise Sheet 10
• Exercise Sheet 11
• Exercise Sheet 12

Literature

A growing file of Mini Lecture Notes:

• ComplexAnalysis2MiniNotes_W13.pdf.

Textbooks:

• Donaldson. Riemann surfaces. Oxford University Press, 2011.
• Springer. Introduction to Riemann surfaces. Addison-Wesley, 1957.
• Lamotke. Riemannsche Flächen. Springer, 2005.
• Weyl. Die Idee der Riemannschen Fläche. Teubner, 1913.
• Miranda. Algebraic Curves and Riemann Surfaces. AMS, 1995.
• Jost. Compact Riemann Surfaces, 3rd ed. Springer, 2006.
• Farkas and Kra. Riemann Surfaces. Springer, 1980.
• Forster. Riemannsche Flächen. Springer, 1977.
• Griffiths and Harris. Principles of Algebraic Geometry. Wiley, 1994.

• McKean and Moll, Elliptic Curves. Cambridge University Press, 1997.
• Kirwan, Complex Algebraic Curves (Chapters 5-6). LMS Student Texts 23, Cambridge 1992,

• Bobenko. Compact Riemann Surfaces. Lecture notes, available on http://page.math.tu-berlin.de/~bobenko/skripte.html
• Donaldson. Riemann Surfaces. Lecture notes, 2004, available on http://www2.imperial.ac.uk/~skdona
• Hitchin. Geometry of Surfaces (Sections 1-3). Lecture notes, 2004, available at http://people.maths.ox.ac.uk/hitchin/hitchinnotes/hitchinnotes.html
• Hitchin. Algebraic Curves (Sections 4-5). Lecture notes, 2009, available at http://people.maths.ox.ac.uk/hitchin/hitchinnotes/hitchinnotes.html

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