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| 65-135: |
Complex functions II -
Modular forms and Riemann surfaces
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| Lecturer: |
Christoph
Schweigert
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| Office hours: |
see
homepage
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| Content: |
1. Elliptic functions
2. Elliptic modular forms
3. Covering spaces and Riemann surfaces
4. Compact Riemann surfaces
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| Goal: |
Upon request, the lectures will be delivered in English.
The goal of this lecture course is to present some topics that extend
and deepen the theory of complex functions as it is presented
in ``Funktionentheorie I'' or classes for physicists.
The two most important topics are:
Modular forms:
To make more explicit the theory of analytic functions, we
introduce and discuss modular forms. These functions enter in fields
as different as number theory, geometry, algebra and representation
theory. They arise as string amplitudes as well.
Riemann surfaces
Problems like defining a complex square root require a careful
rethinking of the domains of analytic functions. The answer is
provided by Riemann surfaces, a theory of much use in itself.
(E.g. the world sheet of a string essentially carries the
structure of a Riemann surface.)
For more information, we refer to
http://www.math.uni-hamburg.de/home/schweigert/ss11/funktionentheorie2.html
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| Requirements: |
The class addresses students in all bachelor and master programs
in mathematics and physics; a knowledge of complex functions
correspondingly roughly to Chapter 1-4 of my
Lecture Notes from Winter 2009/2010 is sufficient.
It is accessible and actually aims at bachelor students with
these prerequesites as well.
Many of the methods have applications in mathematical physics. The class
is suitable for students in the bachelor or master program in physics
and in mathematical physics as well.
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| Literature: |
R. Busam, E. Freitag: Funktionentheorie 1, Springer 2006
auf Englisch als: Complex Analysis, Springer Universitext
O. Forster, Lectures on Riemann surfaces, Springer Graduate Texts in
Mathematics, 1991
M. Koecher, A. Krieg: Elliptische Funktionen und Modulformen, Springer 2007
J.P. Serre: A course in Arithmetic. Springer Graduate Text in Mathematics 7,
1973, see Chapter 7
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| Time and Place: |
Lecture: Tuesday and Friday
14:15-15:45 in Lecture Hall H5. Start on Tuesday,
April 4, 2011.
Sections: Friday 10:15-11:45 in Room 431, Geomatikum.
Start on Friday, April 15.
Additional sections: Wednesday 10:15-11:45
Exceptional schedule:
Lecture instead of sections on Friday,
July 8, in addition to the usual lecture on Friday, July 8 at 2pm.
Sections instead of lecture on Friday, July 15.
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| Exceptional events: |
Tuesday, June 7: dies academicus after 12 am, no lecture.
We agreed that you will read section 4.5 of the notes which will
be only summarized on Friday, June 10.
My notes to be presented as a summary next
Friday are here.
Friday, June 25:
Colloquium in honour of the 100th birthday of Ernst Witt,
no lecture
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| During the term: |
You should solve the problems in close contact with your fellow students,
but write down the solution on your own.
To be admitted to the exam, you should have participated regularly in the
problem sections.
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| Exam: |
Oral exams. Possible Dates
Thursday, August 11, 10-13 and 14 - 17.
Thursday, August 18, 10-13 and 14 - 17.
Thursday, September 8, 10-13 and 14 - 17.
Please fix an appointment with Ms. Dörhöfer at
4 28 38 -5171 or by mail to astrid.doerhoefer@uni-hamburg[dot]de.
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| Lecture notes: |
can be downloaded
here
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| Problem sheets: |
Sheet 1 and
hints
Sheet 2 and
hints
Sheet 3 and
hints
Sheet 4 and
hints
Sheet 5 and
hints
Sheet 6 and
hints
Sheet 7 and
hints
Sheet 8 and
hints
Sheet 9 and
hints
Sheet 10 and
hints
Sheet 11 and
hints
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