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Riemann Surfaces and Complex Analysis (MAST90056)
Graduate courseworkPoints: 12.5Not available in 2024
Overview
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Riemann surfaces arise from complex analysis. They are central in mathematics, appearing in seemingly diverse areas such as differential and algebraic geometry, number theory, integrable systems, statistical mechanics and string theory.
The first part of the subject studies complex analysis. It assumes students have completed a first course in complex analysis so begins with a quick review of analytic functions and Cauchy's theorem, emphasising topological aspects such as the argument principle and Rouche's theorem.
Topics also include: Schwarz's lemma; limits of analytic functions, normal families, Riemann mapping theorem; multiple-valued functions, differential equations and Riemann surfaces. The second part of the subject studies Riemann surfaces and natural objects on them such as holomorphic differentials and quadratic differentials.
Topics may also include: divisors, Riemann-Roch theorem; the moduli space of Riemann surfaces, Teichmueller space; integrable systems.
Intended learning outcomes
After completing this subject, students will gain an understanding of:
- topological aspects of complex analytic functions;
- Riemann mapping theorem and its proof;
- Riemann surfaces;
- holomorphic differentials and line integrals on Riemann surfaces;
- the relevance of this course to further studies in this and related areas.
Generic skills
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- collaborative skills: the ability to work in a team;
- time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 31 January 2024
Eligibility and requirements
Prerequisites
| Code | Name | Teaching period | Credit Points |
|---|---|---|---|
| MAST30021 | Complex Analysis |
Semester 2 (On Campus - Parkville)
Semester 1 (On Campus - Parkville)
|
12.5 |
Or equivalent
Corequisites
None
Non-allowed subjects
None
Inherent requirements (core participation requirements)
The University of Melbourne is committed to providing students with reasonable adjustments to assessment and participation under the Disability Standards for Education (2005), and the Assessment and Results Policy (MPF1326). Students are expected to meet the core participation requirements for their course. These can be viewed under Entry and Participation Requirements for the course outlines in the Handbook.
Further details on how to seek academic adjustments can be found on the Student Equity and Disability Support website: http://services.unimelb.edu.au/student-equity/home
Last updated: 31 January 2024
Assessment
| Description | Timing | Percentage |
|---|---|---|
Assignments | During the teaching period | 60% |
An end-of-semester exam
| During the examination period | 40% |
Additional details
Last updated: 31 January 2024
Dates & times
Not available in 2024
What do these dates mean
Visit this webpage to find out about these key dates, including how they impact on:
- Your tuition fees, academic transcript and statements.
- And for Commonwealth Supported students, your:
- Student Learning Entitlement. This applies to all students enrolled in a Commonwealth Supported Place (CSP).
Subjects withdrawn after the census date (including up to the ‘last day to withdraw without fail’) count toward the Student Learning Entitlement.
Last updated: 31 January 2024
Further information
- Texts
Prescribed texts
None
Recommended texts and other resources
Ahlfors, Lars V. Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978.
Farkas, H. M.; Kra, I. Riemann surfaces. Second edition. Graduate Texts in Mathematics, 71. Springer-Verlag, New York, 1992.
Jost, Jürgen. Compact Riemann surfaces. An introduction to contemporary mathematics. Translated from the German manuscript by R. R. Simha. Universitext. Springer-Verlag, Berlin, 1997.
Lang, Serge. Complex analysis. Fourth edition. Graduate Texts in Mathematics, 103. Springer-Verlag, New York, 1999.
Segal, Sanford L. Nine introductions in complex analysis. Revised edition. North-Holland Mathematics Studies, 208.Elsevier Science B.V., Amsterdam, 2008. - Related Handbook entries
This subject contributes to the following:
Type Name Course Doctor of Philosophy - Engineering Course Ph.D.- Engineering Course Master of Philosophy - Engineering Course Master of Science (Mathematics and Statistics) Informal specialisation Mathematics and Statistics - Available through the Community Access Program
About the Community Access Program (CAP)
This subject is available through the Community Access Program (also called Single Subject Studies) which allows you to enrol in single subjects offered by the University of Melbourne, without the commitment required to complete a whole degree.
Entry requirements including prerequisites may apply. Please refer to the CAP applications page for further information.
- Available to Study Abroad and/or Study Exchange Students
This subject is available to students studying at the University from eligible overseas institutions on exchange and study abroad. Students are required to satisfy any listed requirements, such as pre- and co-requisites, for enrolment in the subject.
Last updated: 31 January 2024