|Introduction to Riemann surfaces|
|Time：||Thu/Fri 13:00-14:50, 2016-07-07 ~ 2016-08-25|
|Instructor：||Lizhen Ji [University of Michigan]|
|Place：||【Updated】Lecture hall, Floor 3, Jin Chun Yuan West Building|
Riemann surfaces have played a fundamental role in mathematics since they were introduced by Riemann in his thesis in 1851. They are 1-dimensional complex manifolds and algebraic curves. Besides their own interest, they are basic ingredients in Teichmuller theory and the theory of moduli spaces. They have also motivated a lot of problems and results on higher dimensional complex manifolds and algebraic varieties.
As Donadlson wrote in the preface to his recent book on Riemann surfaces: “The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry and diverse topics in mathematical physics.”
In this course, we will start from basics and give an introduction to basic results on Riemann surfaces such as the Riemann-Roch Theorem, the uniformization theorem, and some basic results on deformation and moduli spaces of Riemann surfaces, and the geometry of algebraic curves. We will indicate directions of how results on Riemann surfaces can be generalized.
The book by Donaldson will be one of the main references. We will use other books and papers.
Oxford Graduate Texts in Mathematics, 22. Oxford University Press, Oxford, 2011. xiv+286 pp.
2. H. Farkas, I. Kra, Riemann surfaces. Second edition. Graduate Texts in Mathematics, 71. Springer-Verlag, New York, 1992. xvi+363 pp.
Translations of Mathematical Monographs, 76. American Mathematical Society, Providence, RI, 1989. x+221 pp.