Riemann Surfaces & Algebraic Curves - MAT00108H
- Department: Mathematics
- Credit value: 20 credits
- Credit level: H
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Academic year of delivery: 2025-26
- See module specification for other years: 2026-27
Module summary
Riemann surfaces lie at the interface of complex analysis, complex geometry and algebraic geometry. This course begins by developing some more advanced complex analysis before providing 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 important theorems.
Related modules
Pre-requisite modules
Prohibited combinations
Additional information
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.
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Charts and Atlases (such as those studied in Differential Geometry of Curves and Surfaces)
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Distinguishing compact surfaces by their genus (studied in Topology)
The M-level version of this module cannot be taken if H-level was taken.
Module will run
| Occurrence | Teaching period |
|---|---|
| A | Semester 2 2025-26 |
Module aims
Riemann surfaces lie at the interface of complex analysis, complex geometry and algebraic geometry. This course begins by developing some more advanced complex analysis before providing 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 important theorems.
Module learning outcomes
By the end of the course, students will be able to
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Use the implicit function theorem to construct complex charts for a given algebraic curve.
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Construct examples of Riemann surfaces as algebraic curves and compute their genus.
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Integrate a differential form on a Riemann surface.
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Describe the meromorphic functions on a Riemann surface that are bounded below by a given divisor.
Module content
H/M level content.
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Analytic continuation
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Definition of a Riemann surface and examples given by algebraic curves
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Functions on Riemann surfaces and holomorphic maps between Riemann surfaces
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Integration on Riemann surfaces
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Divisors associated to meromorphic functions and one forms
Indicative assessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100.0 |
Special assessment rules
None
Indicative reassessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100.0 |
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.
Indicative reading
Main Text:
Miranda, Algebraic Curves and Riemann Surfaces
Supplementary Texts:
Donaldson, Riemann Surfaces
Kirwan, Algebraic Curves