Required in stage
Optional in stage
First term
Second term
Both terms
Not organised
This year
Alternating years
External
Prerequisites
Taught by
Language of instruction
Duration
Identical coursesRiemann Surfaces (B-KUL-G0B05A)
This course is not taught this academic year, but will be taught next year.Aims
A Riemann surface is a surface on which one can do complex analysis. The study of Riemann surfaces combines techniques from analysis, differential geometry and algebra.
After following this course, the student is familiar with the notion of a Riemann surface, and its connection with algebraic curves. The student is able to learn a new topic by himself and give an exposition about it.
Previous knowledge
Good knowledge of complex analysis in one variable as treated for example in Complexe Analyse (G0O03A).
Is included in these courses of study
Activities
5 ects. Riemann Surfaces (B-KUL-G0B05a)
Content
Basic theory of Riemann surfaces.
(1) Definitions and examples of Riemann surfaces.
(2) Functions on a Riemann surface. Local and global properties.
(3) Integration on a Riemann surface. Holomorphic and meromorphic 1-forms. Residue theorem. Surface integrals and Stokes theorem.
(4) Divisors and meromorphic functions.
(5) Jacobi variety and Abel's theorem.
(6) Riemann-Roch theorem.
Introduction to one advanced topic. Possible topics are:
(7) Jacobi inversion theorem.
(8) Theta functions.
(9) Sheaves and Cech cohomology.
Course material
- Recommended literature: Wilhelm Schlag, A Course in Complex Analysis and Riemann Surfaces, American Mathematical Society, 2014
- Course notes
- Toledo
1 ects. Riemann Surfaces: Exercises (B-KUL-G0B06a)
Content
Basic theory of Riemann surfaces.
(1) Definitions and examples of Riemann surfaces.
(2) Functions on a Riemann surface. Local and global properties.
(3) Integration on a Riemann surface. Holomorphic and meromorphic 1-forms. Residue theorem. Surface integrals and Stokes theorem.
(4) Divisors and meromorphic functions.
(5) Jacobi variety and Abel's theorem.
(6) Riemann-Roch theorem.
Introduction to one advanced topic. Possible topics are:
(7) Jacobi inversion theorem.
(8) Theta functions.
(9) Sheaves and Cech cohomology.
Course material
- Recommended literature: Rick Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics, Vol 5., American Mathematical Society, Providence, RI, 1995.
- Course notes
- Toledo