Weekly outline

  • General

    Riemann surfaces, analytic and algebraic aspects

    Code and credit: MM8032 (Valda ämnen i geometri), 7.5 points.

    Time and place: Tuesdays 13.15-15.00 in room 306, Department of mathematics, Stockholm University, bldn. 6, Kräftriket.

    Lecturers: Jan-Erik Björk (jeb@math.su.se) and Boris Shapiro (shapiro@math.su.se). J-E. Björk will deliver the first 8 lectures and B.Shapiro will deliver lectures 9-12. The remaining 2-3 lectures will be devoted to presentations by the course participants.

    Prerequisites: First course in analytic functions in one complex variable, basic differential geometry in 2 real variables. Familiarity with basic notions of functional analysis, e.g. Hilbert spaces is desirable.

    Examination: Homeworks and/or presentations.

    Basic text: H.Cartan,  Elementary theory of analytic functions of one or several complex variables. Translated from the French.  Dover Publications, Inc., New York, 1995. 228 pp.  P.Griffiths, J.Harris Principles of algebraic geometry. vol 1, John Wiley & Sons, Inc., New York, 1994. xiv+813 pp.  (mainly Ch.2)

    Additional reading: Lecture notes by J-E. Björk will be distributed during the course.

    Timetable and topics:

    September 9. Analytic and harmonic function theory in the complex plane.

    September 16. Continuation.

    September 23. Analysis on compact Riemann surfaces.

    September 30. Continuation.

    October 7.  Uniformization theorem for Riemann surfaces.

    October 14. Algebraic function fields and associated Riemann surfaces.

    October 21. Potential theory, hyperbolic metric and geodesic curves on Riemann surfaces.

    October 28. Aspects of Riemann surfaces via D-modules.

    November 4. Riemann-Hurwitz and Riemann-Roch theorems with applications.

    November 11. Continuation. Abel's theorem.

    November 18. Linear systems on curves.

    November 25. Plücker formulas.

    December 2, 9 and 16. Ph D students presentations.