Riemann Surfaces 2018/19

The lecture is devoted to the introduction to the topics in complex geometry of Riemann surfaces and holomorphic line bundles.

We will cover:

- Branched and unbranched coverings
- Algebraic curves
- Topology of Riemann surfaces
- Abelian forms
- Divisors
- Jacobean variety
- Riemann-Roch theorem
- Abel theorem, Jacobi inversion theorem
- Riemann's theta functions
- Prime forms
- Bosonisation formulas on Riemann surfacess
- Quantum Hall states on Riemann surfaces


The course is structured with the goal to give all the necessary ingredients for proving Fay's bosonisation formulas on Riemann surfaces. The tools that we will learn in this course are also relevant for topics in mathematical physics, such as conformal field theory, string theory and Quantum Hall effect.

Lectures: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10-20, 21-22, 23-24

Homeworks: 0, 1, 2, 3, 4

Classification of surfaces: 1, 2, 3, 4, 5, 6, 7 8, 9, 10, 11
The Classification of Surfaces and the Jordan Curve Theorem

Literature:

[1] J. Jost, "Compact Riemann surfaces: An Introduction to Contemporary Mathematics", Springer, 1997.
[2] X. Ma, G. Marinescu "Holomorphic Morse inequalities and Bergman kernels", Progress in Mathematics v. 254, Birkhauser, 2007.
[3] J. Fay, "Theta functions on Riemann surfaces", Lecture Notes in Mathematics, v. 352, Springer-Verlag, 1973.
[4] H. M. Farkas, I. Kra , "Riemann surfaces", Graduate Texts in Mathematics Volume 71, Springer, 1992.
[5] D. Mumford, "Tata lectures on Theta, I ", Birkhauser, 1983.
[6] A. Bobenko, "Compact Riemann surfaces", lecture notes.
[7] L. Alvarez-Gaume, J.-B. Bost, G. Moore, P. Nelson, C. Vafa, "Bosonization on Higher Genus Riemann Surfaces", Commun. Math. Phys. 112 (1987) 503-552.
[8] S. Klevtsov, "Geometry and Large N limits in Laughlin states", Travaux Math., v. XXIV, (2016) 63-127, arXiv:1608.02928.