COURSE UNIT TITLE

: RIEMANN SURFACES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 5058 RIEMANN SURFACES ELECTIVE 3 0 0 8

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSISTANT PROFESSOR CELAL CEM SARIOĞLU

Offered to

Mathematics (English)
Mathematics (English)

Course Objective

Riemann surfaces can be defined in several different, equivalent ways, for example as one-dimensional complex manifolds, or as oriented two-dimensional real manifolds. In addition, any compact Riemann surface can be embedded in projective space, thus giving it the structure of an algebraic curve. Riemann surfaces therefore appear in many areas of mathematics, from complex analysis, algebraic and differential geometry, to algebraic topology and number theory.

The course will introduce some of the classical results on Riemann surfaces, emphasizing the interplay between topology, complex analysis and geometry. So, we will start with the topological setup and then pass via analysis to algebraic geometry.

Learning Outcomes of the Course Unit

1   will be able to describe what is a Riemann surface.
2   will be able to describe what is a Divisor.
3   will be able to explain the Riemann-Roch theorem.
4   will be able to use the Riemann-Hurwitz formula.
5   will be able to classify the compact Riemann surfaces.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Manifolds, complex manifolds.
2 Riemann Surfaces, examples.
3 Holomorphic functions on a Riemann surface, Holomorphic maps between Riemann surfaces.
4 Meromorphic functions.
5 Fundamental group and covering spaces, Monodromy, The Riemann existence theorem.
6 Topological classification of compact Riemann surfaces.
7 Differential forms, Integration on Riemann Surfaces, the theorems of Stoke s and de Rham.
8 The theorems of Stoke s and de Rham.
9 Degree of a holomorphic map, Divisors, Linear equivalence of divisors.
10 The Riemann-Roch space, Sheaf of meromorphic and holomorphic differential forms.
11 Genus of a compact Riemann surface, Riemann-Roch theorem, Riemann-Hurwitz formula.
12 The Residue map, Serre duality, Applications of the Riemann-Roch theorem.
13 The Jacobian and the Abel-Jacobi map.
14 The Uniformization theorem.

Recomended or Required Reading

Textbook:
1. Hershel M. Farkas, Irwin Kra, Riemann Surfaces, Springer, 1980.
2. Otto Forster, Lectures on Riemann Surfaces, Springer, 1981.
3. Alan F. Beardon, A Primer on Riemann Surfaces, Cambridge University Press, 1984.
4. Simon Donaldson, Riemann Surfaces, Oxford University Press, 2011.
Supplementary Books:
5. Rick Miranda, Algebraic Curves and Riemann Surfaces, American Mathematical Society, 1995.
6. Lars V. Ahlfors, Leo Sario, Riemann Surfaces, Princeton University Press, 2015.
7. Hershel M. Farkas and Irwin Kra, Theta Constants, Riemann Surfaces and the Modular Group, American Mathematical Society, 2001.
References:
8. Wilhelm Schlag, A course in Complex Analysis and Riemann Surfaces, American Mathematical Society, 2014 .
9. Raghavan Narasimhan, Compact Riemann Surfaces, Birkhauser, 1992.
10. Eberhard Freitag, Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Hihger Modular Functions, Springer, 2010.
11. Jürgen Jost, Compact Riemann Surfaces, 3rd edition, Springer, 2006.
12. Ernesto Girondo, Gabino Gonzalez-Diez, Introduction to Compact Riemann Surfaces and Dessin d Enfants, Cambridge University Press, 2012.
13. Henri Paul de Saint-Gervais, Uniformisation des surfaces de Riemann retour sur un théoreme centenaire, ENS editiıns, 2010.
14. Lars V. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, 1979.
15. Hermann Weyl, The concept of a Riemann Surface, 3rd edition, Dover Books, 2009.

Planned Learning Activities and Teaching Methods

Lecture notes, Presentation, Problem solving, Homework Assignments

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 ASG ASSIGNMENT
2 MTE MIDTERM EXAM
3 FIN FINAL EXAM
4 FCG FINAL COURSE GRADE ASG * 0.30 + MTE * 0.30 + FIN * 0.40
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) ASG * 0.30 + MTE * 0.30 + RST * 0.40


Further Notes About Assessment Methods

Midterm exam, Homework Assignments, Final exam

Assessment Criteria

30% (Midterm examination) +30%(Homework assignment)+40% (Final examination)

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

Asst. Prof. Dr. Celal Cem SARIOĞLU
E-mail: celalcem.sarioglu@deu.edu.tr
Phone: +90 232 301 8585
Office: B212 (Mathematics Department)

Office Hours

To be announced later.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 14 3 42
Preparations before/after weekly lectures 13 3 39
Preparation for midterm exam 1 25 25
Preparation for final exam 1 30 30
Preparing assignments 6 8 48
Midterm 1 3 3
Final 1 3 3
TOTAL WORKLOAD (hours) 190

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.1334434343
LO.2334434343
LO.3334434343
LO.4334434343
LO.5334434343