18.116: Riemann Surfaces

This is the course website for the course 18.116 Fall 2016 with material and information relevant to the course.
The class meets 9:00-10:00AM on MWF at 2-136.

In the first part of the course we follow the differential geometric approach presented in the textbook
  • Riemann Surfaces by S. Donaldson, with several additions emphasizing examples and applications.

    Part I. Classical Theory:
    1. Classification of smooth surfaces
    2. Riemann's Existence Theorem
    3. Solution to Poisson's equation
    4. Riemann-Roch Formula
    5. Uniformization Theorem
    6. Abel-Jacobi Theorem
    In the second part of the course we present different topics in which the classical theory of Riemann surfaces has a central role. The material will be based on some of the research articles and textbooks cited below. It is unlikely that we will be able to cover all of them in depth, but we will try.

    Part II. Advanced Topics:
    1. Hyperbolic surfaces, Fuchsian groups
    2. Differential Equations, Riemann-Hilbert
    3. Moduli space of curves, Teichmuller theory
    4. Dessins d'enfants, Belyi's Theorem
    5. Jacobians and Theta functions
    6. Moduli space of vector bundles
    Research articles:
    1. Riemann surfaces and spin structures, M.F. Atiyah, Ann. Sci. de l'ENS (1971).
    2. A new proof of a theorem of Narasimhan and Seshadri, S. Donaldson, J. Diff. Geom. 18 (1983).
    3. The Yang-Mills Equations over Riemann Surfaces, M. F. Atiyah and R. Bott, Phil. Trans. R. Soc. Lond. A (1983).
    4. Linearizing Flows and a Cohomological Interpretation of Lax Equations, Phillip A. Griffiths, American Journal of Mathematics 107 (1985).
    5. Stable bundles and integrable systems, N. Hitchin, Duke Math. J. 54 (1987).
    6. Monodromie des systèmes différentiels linéaires sur la sphère de Riemann, A. Beauville, Sém. Bourbaki 35 (1993).
    7. Dessins d'enfants, J. Oesterlé, Sém. Bourbaki 44 (2001).
    Textbook References:
    1. Curves and their Jacobians, D. Mumford, The University of Michigan Press (1975).
    2. Riemann surfaces, H.M. Farkas, I. Kra, Springer Verlag (1980).
    3. Integrable Systems, N. Hitchin, G. Segal, R. Ward, Oxford Clarendon Press (1999).
    4. Uniformisation des surfaces de Riemann, Henri Paul de Saint Gervais, ENS Éditions (2010).

    Problem Sets
      Problem Set 1: posted on Sep 12, due to Sep 19.

      Problem Set 2: posted on Sep 19, due to Sep 26.

      Problem Set 3: posted on Sep 26, due to Oct 3.

      Problem Set 4: posted on Oct 11, due to Oct 19.

      Problem Set 5: posted on Oct 21, due to Oct 28.

      Midterm Exam: posted on Nov 2, due to Nov 7.

      Problem Set 6: posted on Nov 20, due to Nov 30.

      Final Exam: posted on Dec 15, due to Dec 22.

    Lecture Diaries
    1. Sep 7: Introduction to 18.116.

      Discussion on Elliptic Integrals and Differential Equations as motivation for the study of Riemann Surfaces.
      Solved the motion of a pendulum by using periodicity, leading to Weierstrass rho-function.
      Modern viewpoint on solving elliptic integrals via integration of a non-vanishing holomorphic 1-form.
      Statement of the Uniformization Theorem.

    2. Sep 9: Surface Topology.

      Definition of a smooth surface. Connected sum.
      Statement of the classification of closed surfaces.
      Morse theory: critical points, Morse Lemma.
      Proof of the classification theorem.

    3. Note: In the proof we covered the basic index 0 and index 2 cases, not index 1.
      In the upcoming class of Sep 12 we will complete the proof of the classification.

    4. Sep 12: Surface Constructions.

      Proof of the classification theorem II: index 1 and Tē#RPē=RPē#RPē#RPē.
      Construction of surfaces as quotients of polygons.
      Space of oriented lines. Pairs of points in circle.

    5. Sep 14: Riemann surfaces.

      Definition of a Riemann surface. Maps between Riemann surfaces.
      Automorphisms of the complex plane C and the disk .
      Polygons and upper half plane H: Schwarz maps.
      The modulus is a holomorphic invariant of annuli.
      Holomorphic structure on .

    6. Sep 16: Riemannian surfaces are Riemann surfaces

      Metrics on surfaces. Hyperbolic plane.
      Gauss Theorem. Isothermal coordinates.
      Beltrami Equation. Elliptic coordinates.

    7. Sep 19: Algebraic Curves are Riemann Surfaces

      Implicit Function Theorem. Local algebraic functions.
      Complex projective space. Smooth plane curves.
      The twisted cubic. Projective questions.

    8. Sep 21: Quotient Riemann Surfaces

      C and H quotient by translations.
      H quotient by dilations. C quotient by Zē.
      Genus 2 hyperbolic quotient: hyperbolic octagon.
      Discrete subgroups of PSL(2,R) give Riemann surfaces.
      Fundamental domain for PSL(2,Z).

    9. Sep 26: Basic Properties of Maps

      Local structure of holomorphic maps.
      Proper maps, branch set and degree.
      Riemann surfaces of multi-valued functions.

    10. Sep 28: Riemann Existence Theorem

      Galois correspondence for covering maps.
      Monodromy data and permutation representations.
      Existence of a Riemann Surface from discrete data.

    11. Oct 3: Hyperbolic Geometry Interlude

      Hyperbolic metric on the unit disk. Geodesics.
      Classification of hyperbolic isometries.
      Triangle areas and Euler characteristic.

    12. Oct 5: Normalization of algebraic curves

      Compactification of plane algebraic curves.
      Normalization of singular points.

    13. Oct 7: Riemann-Hurwitz Formula

      Total index of zeroes of meromorphic forms is minus Euler characteristic.
      Riemann-Hurwitz Formula via differential forms.
      Genus-degree formula for plane algebraic curves.

    14. Oct 12: DeRham Cohomology

      Differential Forms on surfaces.
      DeRham Cohomology. Computation for compact surfaces.

    15. Oct 14: Dolbeault Cohomology

      Almost complex structures. Splitting d=∂+.
      Holomorphic and meromorphic forms, (1,0) and (0,1) forms.
      Dolbeault cohomology groups.

    16. Oct 17: Riemann-Roch Formula

      Construction of meromorphic functions with prescribed poles.
      Heuristic dimension count. Riemann-Roch Formula.
      Applications to genus 0, 1 and 2.

    17. Oct 19: Poisson's equation Δu=ρ

      Proof of Riemann-Roch. Residue map and its dual.
      Poisson's Equations and Minimizers.
      Statement of the Dirichlet Principle.

    18. Oct 21: Riesz Representation and Weak Formulation

      Hilbert spaces. Riesz Representation Theorem.
      Weak solutions for Poisson's equation.

    19. Oct 24: Poincaré inequality

      Boundedness of the functional ∫ρ(·).
      Convolutions: inheritance of (the best) integrability.
      Proof of the Poincaré inequality.

    20. Oct 26: Regularity for Lē-weak harmonic functions.

      Green function for the Poisson equation.
      Local smooth solutions Δu=Ρ.
      Mollifiers and regularity.

    21. Oct 28: Elliptic operators

      W1,2 compactness in Lē. Decay Fourier coefficients.
      Fundamental theorem for elliptic PDEs. Hodge theorem.
      Index of an operator. Revisit Riemann-Roch: index count.

    22. Oct 31: Uniformization Theorem

      Statement of the Theorem. Consequences.
      Reduction to existence of decaying meromorphic functions.
      Uniqueness of ends for simply connected surfaces.

    23. Nov 2: 18.116 Midterm

    24. Nov 4: Proof Existence of Decaying Solutions

      Behaviour at infinity: classification.
      Maximum principle and the loop construction.
      Existence of decaying meromorphic functions.

    25. Nov 7: Basics of Sheaves

      Motivation for the definition.
      Restatement of main problems in terms of sheaves.
      Sheaves of global sections and examples.

    26. Nov 9: Sheaf Cohomology

      Cech cohomology. Computations and examples.
      Morphisms of sheaves. Short exact sequences.
      Long exact sequence in cohomology.

    Grade scheme: 0.7·PSets + 0.2·Midterm + 0.1·Final.
    Both the midterm and the final are take-home exams.