SUNY at Stony Brook
MAT 538 Riemann Surfaces Fall 2023

Place and time: MW 2:30pm-3:50pm in Physics P129.

Professor: Leon Takhtajan, Office: Math Tower 5-111.

Office hours: Tu, 2:00pm-4:00pm in 5-111 and by appointment.

Course description: The standard description of MAT 538 is on the course web page. In additon, we will develop various methods for studying Riemann surfaces and rich mathematical structures associated with them. Specifically, we plan (if time permits) to cover the following topics, some of them rather briefly and without proofs.

Complex-analytic & algebro-geometric methods

  • Riemann surfaces as: (i) Riemann srufaces of algebraic functions; (ii) ramified covers of the Riemann sphere; (iii) one-dimensional complex manifolds.
  • Review of sheaves, divisors, line and vector bundles and Hodge theory for Riemann surfaces.
  • Riemann-Hurwitz and Riemann-Roch theorems.
  • Riemann bilinear relations and reciprocity laws for abelian differentials.
  • Abel theorem and Jacobi's inversion theorem.
  • Jacobian variaties, theta-functions and the Riemann theorem.
  • Uniformization theorem and Narasimhan-Seshadri theorem.
  • Introduction to the Krichever's map, Baker-Akhiezer function and the KdV hierarchy.

    • Algebraic methods

  • Algebraic function fields.
  • Differentials and Tate's proof of the residue theorem.

    • Conformal field theory methods

  • Fay's trisecant formula.
  • Bosonization formulas of Alvarez-Gaume, Bost, Moore, Nelson and Vafa.

    Textbook: We will be using variety of sources for different topics and will indicate exact references in a due course.

    Homework:

  • Homework 1.

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    Last updated July 2023