Riemann surfaces and integrable systems

Course for the Master Program in Mathematical Physics, IMB, Fall 2019.

This is a first semester course for the Master Program in Mathematical Physics at the IMB. The lectures will be in English.

Description

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.

Course material

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)

Lectures

  • Multivalued functions on the complex plane and surfaces.
  • The idea of Riemann surface.
  • Definition of Riemann surface: complex chart, atlas, complex structure.
  • The Riemann sphere, $S^2$, and the projective line.
  • Review of the definition of holomorphic function on the complex plane.
  • Complex tori (P).
  • Holomorphic functions on a RS.
  • Riemann's removable singularity theorem (P).
  • Holomorphic maps between Riemann surfaces.
  • Identity theorem for holomorphic maps between Riemann surfaces (P).
  • Isolated singularities of holomorphic functions.
  • The field of meromorphic functions.
  • Meromorphic functions as maps to $\mathbb{C}_\infty$.
  • Order of a meromorphic function at a point.
  • Local form of holomorphic maps between Riemann surfaces (P).
  • 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.
  • Meromorphic functions on $\mathbb{C}_\infty$.
  • Automorphisms of $\mathbb{C}_\infty$.
  • Degree theorem (P).
  • Sum of orders of a meromorphic function over a compact RS.
  • Smooth plane algebraic curves are non-compact Riemann surfaces.
  • Doubly periodic functions and meromorphic functions on the torus.
  • Orientability of Riemann surfaces and classification of 2-manifolds.
  • Triangulations of surfaces and Euler number.
  • Riemann-Hurwitz formula. (P)
  • Some consequences of Hurwitz formula.
  • Holomorphic and meromorphic functions on smooth affine plane curves.
  • Characterisation of ramification points on a smooth affine plane curve.
  • Sheaves. Basic definitons and examples.
  • Stalk of a sheaf.
  • Meromorphic and smooth differentials.
  • Differential of a function and of a 1-form, wedge product.
  • Pull-back of functions and forms. Order of the pull-back.
  • Integration of 1-forms over paths.
  • Integration of 2-forms on 2-chains. Stoke's theorem.
  • Residue of a meromorphic one form at a point. Residue theorem.
  • Divisors. Principal divisors. Degree.
  • Exercise session.
  • Canonical divisors. Examples. Degree of a canonical divisor. Proof.
  • Cech cohomology: cochains, coboundary operator, cocycles and coboundaries.
  • Cech cohomology: interpretation of zeroth and first cohomology groups.
  • Mittag-Leffler problem.
  • Pull back of divisors. Riemann-Hurwitz formula for canonical divisors.
  • Partial order on coverings. Refining maps and induced maps between differential complexes. Homotopy map. Sheaf cohomology as direct limit over the directed set of open coverings.
  • Riemann-Roch theorem.
  • Sheaf of meromorphic functions with poles bounded by a divisor.
  • Index of specialty.
  • Any genus zero Riemann surface is isomorphic to the Riemann sphere.
  • Sheaf of meromorphic 1-forms with poles bounded by a divisor. Sheaf isomorphism.
  • Serre duality.
  • Equivalence of topological, arithmetic, analytic genera.
  • Partition unity and computation of $H^1(X, \mathcal{E})$.
  • Cohomology of locally constant sheaves over a simply connected topological space.
  • Leray coverings. Computation of the first cohomology group of the pointed plane.
  • The skyscraper sheaf and the short exact sequence of sheaves.
  • The long exact sequence in cohomology and the proof of RR theorem.
  • Exercise session.

Exam

Final exam on 9 January 14:00 - 17:00.