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| 65-135: |
Homological algebra
and applications
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| Lecturer: |
Christoph
Schweigert
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| Office hours: |
see
homepage
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| Content: |
Upon request, the lectures will be delivered in English.
1. Simplicial sets and complexes
2. Resolutions
3. Group cohomology and / or Lie algebra cohomology
4. Spectral sequences
5. Derived categories and derived functors
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| Goal: |
The goal of this lecture course is to present some aspects of
homological algebra and its applications. The precise selection of
the material will depend on the background of the audience.
For more information, we refer to
http://www.math.uni-hamburg.de/home/schweigert/ss11/homalg.html
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| Requirements: |
The class addresses students in mathematics and physics who have a
sufficient knowledge of algebra.
You should know roughly the material of Chapter 1 and Chapter
2.1-2.4 of the algebra class, see the
lecture
notes.
Some background reading of Chapter 6 is recommended, but the content of
that chapter will be summarized from a different perspective in the lectures.
Many of the methods have applications in mathematical physics, e.g. in the
BRST and BV quantization. Depending on the background of the students,
we might try to give an elementary introduction or overview to this topics.
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| Seminar: |
Interested students can give a 60 minutes talk, fulling thus the requirement
of a master seminar. The topics are logically independent of the main flow
of the lecture. The talk can be given at any time in the second half of the
term.
Topics:
Homological dimension (Johannes Lederich, Wednesday June 8,
2011, 14:00 in Geom 1241)
References: Weibel, Ch. 4
Explain the global dimension theorem 4.1.2 and the tor-dimension theorem 4.1.3
and their proof. Illustrate rings of low dimension by discussing
theorem 4.2.2. and theorem 4.2.11.
Group cohomology (Kemal Tezgin, Wednesday June 8, 2011, 15:30 in
Geom 1241)
References: Weibel, Ch. 6,
lecture notes by Dietrich Burde, Vienna,
and own
lecture
notes, Ch. 7.
Explain why group cohomology can be computed with the bar complex.
Explain the interpretation of low cohomology groups. The interpretation of
the third cohomology is particularly interesting, but optional.
A
reference where group cohomology is applied to an alternative
understanding of the octonions as an associative algebra.
Lie algebra cohomology (Nils Matthes, Thursday June 9, 2011,
16:00 in Geom 432)
References: Weibel, Ch. 7 and lecture notes by D. Burde, Vienna
Summarize the definition of a Lie algebra and its universal envelopping
algebra. Give interpretations of low cohomology groups (theorem 7.4.1, 7.4.7).
Explain the Chevalley-Eilenberg complex in theorem 7.7.2.
Hochschild cohomology (Steffen Thaysen, June 9, 2011, 17:30
in Geom 432)
References: Weibel, Ch. 9
Explain the complex in section 9.1, explain lemma 9.13. Explain the
interpretations given in 9.2.1, 9.2.2 and 9.3.1.
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| Literature: |
T. Bauer:
Homologische Algebra und Gruppenkohomologie .
Skript zur Vorlesung im WS 2004/2005 an der Universität Münster.
S. Gelfand, Y. Manin: Methods of Homological Algebra, Springer 1997
P.J. Hilton, U. Stammbach: A course in homological algebra, Springer
Graduate Text in Mathematics 4 1997
C. Weibel: An introduction to homological algebra, Cambridge 1995.
T. Chow:
You Could Have Invented Spectral Sequences, Notices of the AMS,
53 (2006) 15-19.
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| Lecture Notes: |
handwritten version as
pdf file
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| Time and Place: |
Lecture: Friday 8:15-9:45
in Lecture Hall H6. Start on Friday, April 8. No lecture on Friday, July 15,
but seminar talks on Wednesday, June 8 and Thursday, June 9 from 14-17.
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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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