Compact Riemann Surfaces (MAGIC006)
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The purpose of the course is to present enough material on compact Riemann surfaces for students to be able to read literature where ideas such as meromorphic differentials, Abel's map and the Jacobi variety, divisor classes and divisor line bundles are used. Compact Riemann surfaces are also the simplest examples of Kaehler manifolds, and every complete smooth algebraic curve is a compact Riemann surface, so they provide an entry into complex manifold theory as well as algebraic geometry. While sheaf theory provides an elegant way of treating many of the topics covered, it will not be explicitly invoked but we will take an approach (and use notation) which is in the spirit of analytic sheaf theory.
Spring 2019 (Monday, January 21 to Friday, March 29)
The approach we take will rely on a good grounding in complex analysis and a little point set topology. Some experience of the differential geometry of surfaces will be helpful.
Riemann surface as a complex manifold (motivated by multi-valued functions); vector fields and differential forms; basics of integration and singular homology for curves on surfaces; the Abel-Jacobi map and Abel's theorem; the Riemann-Roch theorem; (maybe get as far as Weierstrass points).
Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)
The assessment is by take-home examination. The rubric will read: "There are 3 questions on this exam. Please submit solutions to 2 of these. To obtain a pass requires the equivalent of a complete solution to 1 question. This is a pass/fail assessment."
Compact Riemann Surfaces Exam
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