| 65-135: |
Advanced algebra
- homological algebra
(
Deutsche Version)
|
| Lecturer: |
Christoph
Schweigert
|
| Office hours: |
see
homepage
|
| Content: |
1. Moduls over rings
2. Categories, functors and natural transformations
3. Modules over principal ideal domains
4. Representation theory
5. Artinian and noetherian modules
6. Resolutions and derived functors
7. Group cohomology
|
| Goal: |
The goal of this lecture course is to present some algebraic tools used by (almost) all mathematicians. The two most important topics are:
Representation theory:
The goal is to study actions of e.g. groups on vector spaces. The tools we
develop are used whenever systems with symmetries are studied.
Homological algebra
The notion of a module over an algebra generalizes the notion of a vector space
over a field. Compared to linear algebra, however, many new phenomena occur:
for example, it is not true any longer that any submodule has a complement.
Homological algebra provides methods to study such a situation.
Its methods are applied in algebra, combinatorics, geometry and physics.
For more information, we refer to
http://www.math.uni-hamburg.de/home/schweigert/ws10/algebra2.html
|
| Requirements: |
The class addresses students in all bachelor and master programms
in mathematics and physics who have a sufficient knowledge of linear algebra.
It is accessible to bachelor students as well.
Some knowledge about rings (as usually discussed in the bachelors' lecture
on algebra is helpful. The class can be read independently of Algebra I,
provided the student is willing to study some elementary facts on his/her own.
Many of the methods have applications in mathematical physics. The class
is suitable for students in the master program physics and mathematical
physics as well.
|
| Literature: |
J. Jantzen, J. Schwermer: Algebra, Springer 2004.
E-book
on the campus
P.J. Hilton, U. Stammbach: A course in homological algebra, Springer
Graduate Text in Mathematics 4 1997
|
| Time and Place: |
Lecture: Tuesday 12:30-14:00 and Friday
12:15-13:45 in Lecture Hall H4. Start on October 19.
Sections: Friday 14:15-15:45 in Room 432, Geomatikum. Start on October
29.
|
| 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.
|
| Exam: |
Written exam. First exam immediately in February 2011; precise date to be announced.
After the first exam, an oral exam is possible on a voluntary basis in which the result of the written exam can be improved.
Sie sollten die Klausur erfolgreich bearbeiten können, wenn Sie
regelmäßig selbst die Übungsaufgaben bearbeitet haben. Sie
dürfen
keine Unterlagen und Hilfsmittel (Aufzeichnungen, Skript, Bücher,
Taschenrechner) benutzen.
|
| Lecture notes: |
Lecture notes
|
| Problem sheets: |
will be provided
here, as well as
some hints to the solutions. Please register.
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|