id=941   Program
 Introduction to Riemann surfaces and their moduli spaces Student No.： 50 Time： Mon/Tue 13:30-15:05, 2017-07-04~2017-08-29 Instructor： Lizhen Ji  [University of Michigan] Place： Conference Room 1, Floor 1, Jin Chun Yuan West Building Starting Date： 2017-7-4 Ending Date： 2017-8-29

Description:

Riemann surfaces are fundamental objects in mathematics and were introduced by Riemann in his thesis in 1851. They are 1 dimensional complex manifolds and algebraic curves.Moduli spaces of Riemann surfaces and Teichmuller theory have been extensively studied and are still actively studied.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 also indicate some directions of how results on Riemann surfaces can be generalized.

Reference:

1. Simon Donaldson, Riemann surfaces.

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.

3. Phillips Griffiths, Joe Harris,

Principles of algebraic geometry. John Wiley & Sons, Inc., New York, 1994. xiv+813 pp.

4. R. Miranda,  Algebraic curves and Riemann surfaces. Graduate Studies in Mathematics, 5. American Mathematical Society, Providence, RI, 1995. xxii+390 pp.

5. Joe Harris, Ian Morrison, Moduli of curves, Springer-Verlag, New York, 1998. xiv+366 pp.

6. Henri Paul de Saint-Gervais, Uniformization of Riemann surfaces, European Mathematical Society (EMS), Zürich, 2016. xxx+482 pp.