Module Details |
| The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module. |
| Title | RIEMANN SURFACES | ||
| Code | MATH340 | ||
| Coordinator |
Dr T Boiko Mathematical Sciences T.Boiko@liverpool.ac.uk |
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| Year | CATS Level | Semester | CATS Value |
| Session 2016-17 | Level 6 FHEQ | Second Semester | 15 |
Aims |
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To introduce to a beautiful theory at the core of modern mathematics. Students will learn how to handle some abstract geometric notions from an elementary point of view that relies on the theory of holomorphic functions. This will provide those who aim to continue their studies in mathematics with an invaluable source of examples, and those who plan to leave the subject with the example of a modern axiomatic mathematical theory. |
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Learning Outcomes |
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Students should be familiar with the most basic examples of Riemann surfaces: the Riemann sphere, hyperelliptic Riemann surfaces, and smooth plane algebraic curves. |
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Students should understand and be able to use the abstract notions used to build the theory: holomorphic maps, meromorphic differentials, residues and integrals, Euler characteristic and genus. |
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Syllabus |
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| 1 |
-A brief introduction to topology and topological spaces. -Some topological properties: connected, path-connected, sequentially compact and Hausdorff. -The definition of a Riemann surface. Charts and atlas. Examples. -Plane algebraic curves that are also Riemann surfaces. A weak form of Bezout''s theorem. -More advanced complex analysis material. Identity principle for holomorpic maps. Open mapping theorem. Proof of the holomorphic implicit function theorem in 2 variables. -Maps of Riemann su
rfaces. Isomorphisms of Riemann surfaces. Canonical domains and their automorphisms. -Holomorphic and meromorphic differentials on Riemann surfaces. -Residues and integrals of meromorphic differentials. The residue theorem for compact Riemann surfaces. -Finite triangulation, Euler characteristic and genus. The topological classification of compact Riemann surfaces. -Riemann-Hurwitz formula and its consequences. -The Riemann-Roch theorem. -Uniformisation of Riemann surfaces.
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Recommended Texts |
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Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module. Explanation of Reading List: The core of this module is based on the first chapter of the book of Girondo and Gonzalez-Diez, "Introduction to compact Riemann surfaces and dessins d''enfants". The lectures of the first two weeks are taken from the book of Sutherland, "Introduction to metric and topological spaces". The discussion on smooth plane algebraic curves is taken from the book of Kirwan, "Complex algebraic curves". The discussion on Riemann-Roch is taken from Prof. Nikulin''s online notes. |
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Pre-requisites before taking this module (other modules and/or general educational/academic requirements): |
| MATH243; MATH101; MATH102; MATH103 Some acquaintance with MATH241 would help but is not required |
Co-requisite modules: |
Modules for which this module is a pre-requisite: |
Programme(s) (including Year of Study) to which this module is available on a required basis: |
Programme(s) (including Year of Study) to which this module is available on an optional basis: |
| Programme:G100 Year:3 Programme:G101 Year:3,4 Programme:G110 Year:3 Programme:G1F7 Year:3 Programme:G1N2 Year:3 Programme:G1R9 Year:4 Programme:G1X3 Year:3 Programme:GG13 Year:3 Programme:GN11 Year:3 Programme:GG14 Year:3 Programme:GL11 Year:3 Programme:GR11 Year:4 Programme:GV15 Year:3 Programme:BCG0 Year:3 Programme:L000 Year:3 Programme:Y001 Year:3 Programme:MMAS Year:1 |
Assessment |
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| EXAM | Duration | Timing (Semester) |
% of final mark |
Resit/resubmission opportunity |
Penalty for late submission |
Notes |
| Unseen Written Exam | 2.5 hours | Second semester | 100 | Yes | Standard UoL penalty applies | Assessment 1 Notes (applying to all assessments) Candidates may attempt all questions. The best FIVE answers will be taken into account. Each question carries the same weight. |
| CONTINUOUS | Duration | Timing (Semester) |
% of final mark |
Resit/resubmission opportunity |
Penalty for late submission |
Notes |