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 - 16:00, Wed 16:00 - 18:00. The lectures will take place on Tuesday in Room René Baire (fourth floor of the A building) and on Wednesday in Room A 107 (first floor). For a more detailed calendar see below. The final evaluation will take place in November.
The course will be an introduction to Riemann surfaces and to some of their relations with integrable systems. We will start by reviewing some known fact complex analysis and considering some motivation for introducing Riemann surfaces. Then we will develop some of the classical theory of Riemann surfaces. If time permits we will consider some application to the theory of integrable systems, for example the construction of 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, divisors, Abel theorem, Riemann-Roch theorem.
We will mostly refer to the following books:
[For] O. Forster. Lectures on Riemann surfaces. (1981)
[Mir] R. Miranda. Algebraic curves and Riemann surfaces. (1995)
1. 19/9 16:00 A 107
(0) Example of multivalued function: the square root. Different definitions of holomorphic functions.
(1) Definition of Riemann surface: two dimensional manifolds, complex chart, holomorphic compatibility, maximal atlas, complex structure.
(2) Examples of RS structures: the complex plane, the Riemann sphere, domains, the torus.
2. 21/9 14:00 Baire
(3) Holomorphic functions on Riemann surfaces.
(4) Riemann's removable singularity theorem. Isolated singularities of holomorphic functions.
(5) Holomorphic maps between Riemann surfaces.
(6) Identity theorem for holomorphic maps between Riemann surfaces (proof).
3. 25/9 14:00 Baire
(7) The field of meromorphic functions on a RS, meromorphic functions as maps to $\mathbb{P}^1$. (8) Local form of holomorphic maps between Riemann surfaces (proof).
Exercises 1 and 2 due.
4. 26/9 16:00 A 107
(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, maximum principle. (10) Meromorphic functions on $\mathbb{P}^1$. Doubly periodic functions and meromorphic functions on the torus. (11) Holomorphic maps between compact Riemann surfaces and ramified coverings (proof).
Exercise 3 due.
5. 2/10 14:00 Baire
Order of a meromorphic function at a point. (12) Smooth plane algebraic curves are non-compact Riemann surfaces.
Exercises 4, 5, 6 due.
6. 3/10 16:00 A 107
(12a) Sheaves. Stalk of a presheaf at a point. (*) Exercises.
Exercise 7 due.
7. 9/10 14:00 Baire
(12b) Mittag-Leffler problem, introduction. (12c) Cech cohomology: cochains, coboundary operator, cocycles and coboundaries, interpretation of zeroth and first cohomology groups. Partial order on coverings.
8. 18/10 10:15 A 107
(12d) Refining maps and induced maps between differential complexes. Homotopy map. Sheaf cohomology as direct limit over the directed set of open coverings. (13) Orientability of Riemann surfaces and classification of 2-manifolds.
Exercise 8 due.
9. 19/10 10:00 Baire
Triangulations of surfaces and Euler number. Riemann-Hurwitz formula. (14) Holomorphic differentials.
10. 23/10 14:00 Baire
Meromorphic and smooth differentials. Differential of a function and of a 1-form, wedge product.
11. 24/10 16:00 A 107
Pull back of functions and differentials. Relation between the order of a 1-form and the order of its pull-back. (16) Poincaré lemma, Dolbeault's lemma. (17) Paths. Integration along paths.
12. 6/11 14:00 Baire
(19) Integration of 2-forms on 2-chains. Stoke's theorem. (20) Residue of a meromorphic one form at a point. Residue theorem. (21) Homotopy, fundamental group.
13. 8/11 14:00 A 108
(22) Divisors. Principal divisors, canonical divisors. Examples.
(23) Degree of a canonical divisor. Proof.
14. 13/11 14:00 Baire
(24) Pull back of divisors. Riemann-Hurwitz formula for canonical divisors.
(27) Riemann-Roch theorem. Any genus zero Riemann surface is isomorphic to the Riemann sphere.
15. 15/11 14:00 A 107
(28) Sheaf of meromorphic 1-forms with poles bounded by a divisor. Sheaf isomorphism.
(29) Serre duality. (30) Equivalence of topological, arithmetic, analytic genera.
(30) Review of cohomology. Cohomology of smooth sheaves. Cohomology of locally constant sheaves over a simply connected topological space. Leray coverings. Computation of the first cohomology group of the pointed plane.
16. 20/11 14:00 Baire
(31) More on Dolbeault Lemma and finite dimension of $H^1(X, \mathcal{O})$. (32) From short exact sequences of sheaves to long exact sequences in cohomology. (33) The skyscraper sheaf, the related short exact sequence, and the proof of the Riemann-Roch theorem.
Reading sessions will take place in Room 214 on Thursday at 10:00 - 12:00.
The plan for the reading sessions is the following:
25/10 Klaydson + Florian
1/11 Albin + Nischal
8/11 Helal + Pierre-Alexandre
15/11 Drhuv + Sandhya
22/11 Chaabane + Taiwo
Final exam on 9 January 9:00 - 12:00 A 203.