The notion of a Riemann surface. Basic examples. Holomorphic and meromorphic functions and maps. Topology of Riemann surfaces. Covering spaces and deck transformations. Analytic continuation. Algebraic functions. Integration on Riemann surfaces. Riemann surfaces as complex curves. Basics of sheaf theory.

Construction of meromorphic functions by L2-method. Weyl lemma. Hilbert space of square integrable forms. Meromorphic functions and differentials. Harmonic and analytic differentials. Bilinear relations. Divisors and holomorphic line bundles. The Riemannov-Roch theorem and applications.

Other possible topics: Open Riemann surfaces. The Dirichlet problem. The Runge approximation theorem. Theorems of Mittag-Leffler and Weierstrass. Riemann-Koebe uniformization theorem. Riemann-Hilbert boundary value problem. Serre duality. Abel's theorem and applications. Jacobi inverse problem. Complex tori. Elliptic functions. Weierstrass function.

# Riemann surfaces

H. M. Farkas, I. Kra: Riemann Surfaces, 2nd edition, Springer, New York, 1992.

O. Forster: Lectures on Riemann Surfaces, Springer, New York, 1999.

F. Kirwan: Complex Algebraic Curves, Cambridge Univ. Press, Cambridge, 1992.

B. A. Dubrovin, A. T. Fomenko, S. P. Novikov: Modern Geometry - Methods and Applications III : Introduction to Homology Theory, Springer, New York, 1990.

D. Varolin: Riemann surfaces by way of complex analytic geometry. Amer. Math. Soc., Providence, RI, 2011.

Students learns some of the basic concepts and methods of the theory of Riemann surfaces and its connection to related fields of mathematics such as complex analysis and algebraic geometry. Basic methods of analysis, algebra and topology are applied in the course.

With individual presentations and team work interactions within seminar/project activities students acquire communication and social competences for successful team work and knowledge transfer.

Knowledge and understanding: Undestanding of fundamental topics in the theory of Riemann surfaces.

Application: Riemann surfaces appear naturally in many areas of mathematics (e.g. in analytic and algebraic geometry, differential geometry, symplectic geometry and other areas), as well as in several areas of physiscs (such as string theory) and in other sciences. Elliptic curves are a fundamental tool in cryptography.

Reflection: Understanding the theory on the basis of examples. Acquiring skills in applying the theory to diverse scientific problems.

Transferable skills: The ability to identify, formulate and solve scientific problems using methods of Riemann surface theory. Developing skills of using the domestic and foreign literature. Developing skills of independent presentation of the material.

Lectures, seminar presentations, exercises, homeworks, consultations

Type (homework, seminar paper, oral exam):

homework and seminar paper

oral exam

Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Miran Černe:

ČERNE, Miran. Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces. American journal of mathematics, ISSN 0002-9327, 2004, vol. 126, no. 1, str. 65-87. [COBISS-SI-ID 12895577]

ČERNE, Miran, FORSTNERIČ, Franc. Embedding some bordered Riemann surfaces in the affine plane. Mathematical research letters, ISSN 1073-2780, 2002, vol. 9, no. 5-6, str. 683-696. [COBISS-SI-ID 12391257]

ČERNE, Miran, FLORES, Manuel. Quasilinear [overline{partial}]-equation on bordered Riemann surfaces. Mathematische Annalen, ISSN 0025-5831, 2006, vol. 335, no. 2, str. 379-403. [COBISS-SI-ID 13970777]

Barbara Drinovec Drnovšek:

DRINOVEC-DRNOVŠEK, Barbara. Discs in Stein manifolds containing given discrete sets. Mathematische Zeitschrift, ISSN 0025-5874, 2002, vol. 239, no. 4, str. 683-702. [COBISS-SI-ID 11567449]

DRINOVEC-DRNOVŠEK, Barbara. Proper discs in Stein manifolds avoiding complete pluripolar sets. Mathematical research letters, ISSN 1073-2780, 2004, vol. 11, no. 5-6, str. 575-581. [COBISS-SI-ID 13311065]

DRINOVEC-DRNOVŠEK, Barbara, FORSTNERIČ, Franc. Holomorphic curves in complex spaces. Duke mathematical journal, ISSN 0012-7094, 2007, vol. 139, no. 2, str. 203-254. [COBISS-SI-ID 14351705]

Franc Forstnerič:

FORSTNERIČ, Franc. Runge approximation on convex sets implies the Oka property. Annals of mathematics, ISSN 0003-486X, 2006, vol. 163, no. 2, str. 689-707. [COBISS-SI-ID 13908825]

FORSTNERIČ, Franc, WOLD, Erlend Fornæss. Bordered Riemann surfaces in C [sup] 2. Journal de Mathématiques Pures et Appliquées, ISSN 0021-7824. [Print ed.], 2009, vol. 91, issue 1, str. 100-114. [COBISS-SI-ID 15395417]

FORSTNERIČ, Franc, WOLD, Erlend Fornæss. Embeddings of infinitely connected planar domains into C [sup] 2. Analysis & PDE, ISSN 2157-5045, 2013, vol. 6, no. 2, str. 499-514. [COBISS-SI-ID 16645209]

Pavle Saksida:

SAKSIDA, Pavle. Maxwell-Bloch equations, C Neumann system and Kaluza-Klein theory. Journal of physics. A, Mathematical and general, ISSN 0305-4470, 2005, vol. 38, no. 48, str. 10321-10344. [COBISS-SI-ID 13802073]

SAKSIDA, Pavle. Lattices of Neumann oscillators and Maxwell-Bloch equations. Nonlinearity, ISSN 0951-7715, 2006, vol. 19, no. 3, str. 747-768. [COBISS-SI-ID 13932377]

SAKSIDA, Pavle. Integrable anharmonic oscillators on spheres and hyperbolic spaces. Nonlinearity, ISSN 0951-7715, 2001, vol. 14, no. 5, str. 977-994. [COBISS-SI-ID 10942809]