



Riemann Surfaces 2018/19
Semyon Klevtsov and George Marinescu
Di. 16:00  17:30, Hörsaal (Raum 203),
Do. 10:00  11:30, Seminarraum 1
There will be an extra lecture on the 8.01.2019 at 10 am in
CohnVossen Raum
A test takes place on the 29.01.2019 at 2 pm in Übungsraum 1 (Raum 119)

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
 RiemannRoch 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,
1020,
2122,
2324
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, SpringerVerlag, 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. AlvarezGaume, J.B. Bost, G. Moore, P. Nelson, C. Vafa, "Bosonization on Higher Genus Riemann Surfaces", Commun. Math. Phys. 112 (1987) 503552.
[8] S. Klevtsov, "Geometry and Large N limits in Laughlin states", Travaux Math., v. XXIV, (2016) 63127, arXiv:1608.02928.


