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Module Leader: Ruadhaí Dervan
The theory of Riemann surfaces lies between differential geometry, algebraic geometry, and complex analysis. Riemann surfaces can be viewed as the simplest examples of complex manifolds or (when compact) algebraic varieties, as they are one-dimensional versions of these heavily studied higher-dimensional objects. This course will take a modern approach to Riemann surfaces, and will use the setting of Riemann surfaces to develop many important tools which have become essential in higher-dimensional geometry, with an emphasis on techniques that bridge algebraic and differential geometry. Thus the course is aimed at anyone who would like to gain an appreciation for these ideas, which should be useful to anyone with an interest in geometry (differential or algebraic) or analysis.
The course will follow Donaldson's book "Riemann surfaces" quite closely. Very similar lecture notes are available online at this URL. In particular, we will aim to cover:
- the basic definitions and examples of Riemann surfaces;
- holomorphic maps between Riemann surfaces and the relation with topology;
- calculus on manifolds as it applies to Riemann surfaces;
- the existence of meromorphic functions on Riemann surfaces and Riemann-Roch;
- the uniformisation theorem;
- line bundles, compact Riemann surfaces are projective varieties.
The main results of the course will be the final three bullet points, which demonstrate clearly the links with algebraic geometry. The proofs of these results will involve some linear PDE theory which will be developed in this part of the course, which also provide a foundation for the many related results in geometry that we will not touch upon.
The course will be taught in person for students in Glasgow (and streamed online for those elsewhere). If you would like to attend virtually but are not enrolled in the course (for example because you are not a first-year PhD student in Scotland), please email me and I'll send you information about this. The lectures will be 10:00-12:00 on Tuesdays, and the lecture theatre will vary from week to week:
10th October: 12 UNIV GDNS:101
17th October: ST ANDREWS BUILDING:ROOM 345
24th October: 12 UNIV GDNS:101
31st October: 12 UNIV GDNS:101
7th November: ST ANDREWS BUILDING:ROOM 368 SEM
14th November: 12 UNIV GDNS:101
21st November: ST ANDREWS BUILDING:ROOM 368 SEM
28th November: ST ANDREWS BUILDING:ROOM 345