This is a first semester course for the Master Program in Mathematical Physics at the IMB. The lectures will be in English. The schedule is: Tue 14:00 - 15:30, Thu 10:30 - 12:00. For a more detailed calendar see below. The lectures will take place in Room René Baire (fourth floor of the A building) and in Room A E01 (at the top of the stairs in front of Amphitheater Niepce). The evaluation will take the form of written exam in January.
The course will be an introduction to Riemann surfaces and to some of their relations with integrable systems. We will start with an introduction to classical integrable systems: review of classical Hamiltonian mechanics, Poisson manifolds, Liouville-Arnold theorem. Later we will focus on Riemann surfaces, with the (ideal) aim of constructing solutions to the KP hierarchy via the Baker-Akhiezer functions. Topics will include: definition of Riemann surface and basic examples, plane algebraic curves, hyperelliptic curves, holomorphic coverings, fundamental group, Riemann-Hurwitz theorem, homology groups, abelian differentials, meromorphic functions, Abel theorem, Riemann-Roch theorem.
Reading sessions with seminars and discussions by and for the students of this course will take place in Room 214 each Wednesday at 14:00-16:00. The aim is to review some basic preliminary material for the course and to discuss lecture notes and problems.
The plan for the reading sessions is the following:
1. 26/9 14:00-15:30 (Room A E01): Introduction to finite dimensional integrable systems. Hamilton equations, Poisson brackets, first integrals. Poisson and symplectic manifolds. Completely integrable systems, Liouville-Arnold theorem, action-angle coordinates.
Some optional reading on the subjects treated today:
2. 28/9 10:00-11:30 (Room A E01): (0) Example of multivalued function: the square root. (1) Definition of Riemann surface: two dimensional manifolds, complex chart, holomorphic compatibility, maximal atlas, complex structure. (2) Examples of comples structures: The complex plane, the Riemann sphere, domains, the torus. (3) Holomorphic functions on Riemann surfaces.
3. 3/10 14:00-15:30 (Room René Baire): (4) Riemann's removable singularity theorem. (5) Holomorphic maps between Riemann surfaces.
4. 5/10 10:30-12:00 (Room A E01): (6) Identity theorem for holomorphic maps between Riemann surfaces (and for holomorphic functions on $\mathbb{C}$). (7) The field of meromorphic functions on a RS, meromorphic functions as maps to $\mathbb{P}^1$.
5. 10/10 14:00-15:30 (Room René Baire): (8) Local structure of holomorphic maps between Riemann surfaces. (9) Corollaries: open mapping theorem, inverse function theorem, good behaviour almost everywhere, maps over compact surfaces, holomorphic functions on compact surfaces are constant, Liouville theorem, fundamental theorem of algebra.
6. 12/10 10:30-12:00 (Room A E01): (cont.) Maximum principle. (10) Meromorphic functions on $\mathbb{P}^1$.
7. 17/10 14:00-15:30 (Room René Baire): (cont.) Doubly periodic functions and meromorphic functions on the torus. (11) Holomorphic maps between compact Riemann surfaces and ramified coverings. (12) Smooth plane algebraic curves.
8. 19/10 10:30-12:00 (Room A E01): (cont.) Smooth plane algebraic curves are non-compact Riemann surfaces. (13) Genus of a surface. Triangulations of surfaces and Euler number. Riemann-Hurwitz formula.
9. 24/10 14:00-15:30 (Room René Baire): (cont.) Proof of Riemann-Hurwitz formula. (14) Holomorphic differentials.
10. 9/11 10:30-12:00 (Room A E01): (cont.) Meromorphic and smooth differentials. Exercises.
11. 14/11 14:00-15:30 (Room René Baire): (15) Differential of a function and of a 1-form, wedge product, pull back. Relation between the order of a 1-form and the order of the pull-back.
12. 16/11 10:30-12:00 (Room A E01): (16) Poincaré lemma, Dolbeault's lemma. (17) Paths. Integration along paths.
13. 21/11 14:00-15:30 (Room René Baire): (cont.) Integration along 1-chains. (18) Differentiable singular p-simplexes. p-chains. Boundary operator. p-cycles, p-boundaries. Homology groups. (19) Integration of 2-forms on 2-chains. Stoke's theorem.
14. 23/11 10:30-12:00 (Room A E01): (20) Residue of a meromorphic one form at a point. Exercises, review.
15. 28/11 14:00-15:30 (Room René Baire): (cont.) Residue theorem. (21) Homotopy, fundamental group. (22) Divisors.
16. 30/11 10:30-12:00 (Room A E01): (cont.) Principal divisors, canonical divisors. Examples. (23) Degree of a canonical divisor. Proof.
17. 1/12 16:00-17:30 (Room René Baire): (24) Pull back of divisors. Riemann-Hurwitz formula for canonical divisors.
18. 12/12 14:00-15:30 (Room René Baire): (25) Linear equivalence. (26) $L(D)$
19. 14/12 10:30-12:00 (Room A E01): (cont.) Complete linear system of divisors. $L^{(1)}(D)$. Isomorphisms between $L(D)$ and $L^{(1)}(D)$ spaces. Computation of $L(D)$ for the Riemann sphere. (27) Riemann-Roch theorem. Any genus zero Riemann surface is isomorphic to the Riemann sphere.
20. 19/12 14:00-15:30 (Room René Baire): (28) Existence of meromorphic functions on compact Riemann surfaces. Laurent approximation theorem. Every Riemann surface can be holomorphically embedded in a projective space.
The oral exams (30-40 min) will take place on Friday 19/01/2018 14:00 at my office 404. Roughly four questions (definitions, explanations, proofs, or exercises) will be asked from the syllabus above.
Some literature we might be referring to, to be updated during the course.
[AS] L. V. Alfhors, L. Sario. Riemann surfaces. (1960)
[Arn] V. I. Arnold. Mathematical Methods of Classical Mechanics. (1989)
[Bab] O. Babelon, D. Bernard, M. Talon. Introduction to classical integrable systems. (2003)
[Bob] A. I. Bobenko. Introduction to Compact Riemann Surfaces. In Computational Approach to Riemann Surfaces. Lecture Notes in Mathematics 2013. (2011)
[Dub] B. Dubrovin. Integrable systems and Riemann surfaces, lecture notes. (2009) pdf
[Dun] M. Dunajski. Integrable systems, lecture notes. (2012) pdf
[FK] H. M. Farkas, I. Kra. Riemann surfaces. (1980)
[For] O. Forster. Lectures on Riemann surfaces. (1981)
[Jan] K. Jänich. Topology. (1984)
[Mir] R. Miranda. Algebraic curves and Riemann surfaces. (1995)
[Spr] G. Springer. Introduction to Riemann surfaces. (1957)
[Tel] C. Teleman. Lecture notes on Riemann surfaces. (2003) pdf
[War] F. W. Warner. Foundations of Differentiable Manifolds and Lie Groups. (1983)
[Wey] H. Weyl. The concept of a Riemann surface. (1964)