Prerequisites:
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* Basic Algebra (Linear Algebra, Groups, Rings and Fields, e.g. Chapters I-IV and XIII in S. Lang's book 'Algebra', Springer GTM 211)
* Basic point set topology, ideally including a first introduction to the fundamental group
* Familiarity with the notion of a manifold
* Complex Analysis: holomorphic and meromorphic functions in one variable, and Cauchy's Integral Theorem.
Aim of the course:
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In this course we introduce and study Riemann surfaces, which are 1-dimensional complex manifolds. The study of Riemann Surfaces is a fascinating area of mathematics which mixes ideas from geometry, algebra, analysis and topology. We will see how to naturally generalise many notions and results from the complex plane to Riemann surfaces, such as holomorphic and meromorphic functions. We will study the theory through many concrete examples, such as hyperelliptic Riemann surfaces.
The main goal of the course is to describe the geometry of compact Riemann surfaces. Among many interesting theorems we will cover, one of the most important in terms of its wide-ranging geometric consequences is the Riemann-Roch Theorem. This will enable us to study projective embeddings of compact Riemann surfaces and remarkably allows us to relate them to complex algebraic curves.
Rules about homework and examination:
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During the semester, there will be three homework assignments. The average grade of these assignments counts for 25% of the final grade (also in the case of a resit). At the end of the semester, there will be a written exam. In order to pass the course, the grade of the final exam has to be 5 or higher, regardless of your grade for the homework assignments.
Literature:
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The book that is closest in spirit to what we will cover is Miranda's book 'Algebraic Curves and Riemann Surfaces' (Graduate Studies in Mathematics, Volume 5, AMS). We will also provide written notes for the lectures.
Here is a list of suggested further reading. (This is not required reading material, but these books have a different emphasis from our course and provide complementary points of view.)
* O. Forster, 'Lectures on Riemann Surfaces' (Graduate Texts in Mathematics, Volume 81). The approach of this book is more analytic in nature and in particular here you can find a proof of the existence of enough meromorphic functions on compact Riemann surfaces (we will state and use this fact without proof in the course, as its proof involves a lengthy argument in functional analysis that would not leave us time to discuss the beautiful geometric consequences).
* S. Donaldson, 'Riemann Surfaces' (Oxford Graduate Texts in Mathematics, Volume 22). Here is a second more analytic reference, which also has a proof of the existence of enough meromorphic functions and then goes on to prove the uniformization theorem.
* D. Huybrechts, 'Complex Geometry – an Introduction' (Universitext). This is a book dealing with higher dimensional complex manifolds, with a focus on their geometry. Especially the first chapter on complex manifolds is highly recommended as supplementary reading material for our course.
Lecturers:
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Ariyan Javanpeykar
- Docent: Ariyan Javan Peykar