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# Riemann Surfaces & Algebraic Curves - MAT00111M

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• Department: Mathematics
• Module co-ordinator: Dr. Graeme Wilkin
• Credit value: 20 credits
• Credit level: M
• Academic year of delivery: 2023-24

## Module summary

Riemann surfaces lie at the interface of complex analysis, complex geometry and algebraic geometry. This course provides an introduction to Riemann surfaces, from the initial concepts through to more advanced applications. The course will emphasise concrete examples of algebraic curves to help visualise the theoretical concepts.

## Related modules

• None

### Prohibited combinations

• None

A course in complex analysis that covers holomorphic and meromorphic functions, Laurent series, complex integration and Cauchy’s theorem will give sufficient background for this course. It is expected that students have sound knowledge of all of these topics before the course begins.

Some of the concepts used in this course will be introduced at a fast pace, therefore it will be useful to have seen the topics below.

• Charts and Atlases (such as those studied in Differential Geometry of Curves and Surfaces)

• Distinguishing compact surfaces by their genus (studied in Topology)

## Module will run

Occurrence Teaching cycle
A Semester 2 2023-24

## Module aims

Riemann surfaces lie at the interface of complex analysis, complex geometry and algebraic geometry. This course provides an introduction to Riemann surfaces, from the initial concepts through to more advanced applications. The course will emphasise concrete examples of algebraic curves to help visualise the theoretical concepts.

## Module learning outcomes

By the end of the course, students will be able to

1. Use charts and atlases to prove that a given system of equations defines a Riemann surface.

2. Construct examples of Riemann surfaces as algebraic curves and compute their genus.

3. Integrate a differential form on a Riemann surface.

4. Use the long exact sequence to compute sheaf cohomology groups.

5. Use the Riemann-Roch theorem to compute sheaf cohomology groups.

## Module content

• Basic definitions and examples

• Functions on Riemann surfaces and holomorphic maps between Riemann surfaces

• More examples of Riemann surfaces

• Integration on Riemann surfaces

• Divisors and maps to projective space

• Introduction to sheaves

• Riemann-Roch, Serre duality and applications

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Riemann Surfaces & Algebraic Curves
3 hours 80
Essay/coursework
Coursework 1
N/A 5
Essay/coursework
Coursework 2
N/A 5
Essay/coursework
Coursework 3
N/A 5
Essay/coursework
Coursework 4
N/A 5

### Special assessment rules

None

• Four homework sets (5% each) which are separate from (but related to) the associated seminar (first four seminars). Extensions are possible, as solutions won’t be released until all the marking is complete. [The fifth seminar will have a formative assignment associated with it].

• Final exam (80%).

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Riemann Surfaces & Algebraic Curves
3 hours 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.