Math 234, Riemann Surfaces, Winter 2018

• Lectures: TTh 1:30-3:05pm, McHenry 1279

• Suggested Texts: Riemann Surfaces by Simon Donaldson, Oxford Graduate Texts in Mathematics, 22, Oxford University Press, 2013 and Lectures on Riemann Surfaces by Otto Forster, Graduate Texts in Mathematics, 81, Springer-Verlag New York, 1981.

• Preparation requirements: active knowledge of basic analysis and in particular complex analysis; some knowledge of manifolds and differential geometry.

• Instructor: Viktor Ginzburg; office: McHenry 4124
  email: ginzburg(at)ucsc.edu, phone: 459-2218

• Office Hours: Tu 12:30-1:30pm or by appointment

• Tentative Syllabus: Riemann surfaces are among the most important objects in mathematics. They are ubiquitous and arise in virtually all fields of mathematics and many areas for mathematical physics from analysis and complex analysis to differential equations to number theory and Galois theory to algebraic and differential geometry to string theory and mirror symmetry. One cannot imagine modern mathematics without Riemann surfaces. In fact, one could probably argue that the theory of Riemann surfaces is one of the starting points of modern math.

  In this course, moving at a relaxed pace, we will cover the basics of Riemann surfaces: examples, constructions, the origins, basic properties, connections with topology of surfaces. In the second part of the class, if time permits, we will look into some deeper results and more advanced topics such as the uniformization theorem, the Riemann-Roch theorem, and applications of Riemann surfaces to symplectic topology.