65-405:  Advanced algebra - homological algebra and representation theory
Lecturer:
Exercises:
Office hours: see homepage
Content: 1. Modules 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.
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 Texts in Mathematics 4, 1997
S. Lang: Algebra. Springer Graduate Texts in Mathematics 211, 2005
Time and Place: Lecture: Tuesday 12:15-13:45 and Thursday 12:00-13:30 in Lecture Hall H4. Start on April 4. Second lecture on Thursday, April 6. Then hopefull shift to Wednesday.
Sections: Thursday, 8:15-9:45 in Room 434, Geomatikum. Start on April 6.
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: Individual oral exam in English or German, by individual appointment. Possible dates in July and September 2017 will be announced.
Lecture notes: Lecture notes
Problem sheets: are provided here, as well as some hints to the solutions. Please register in Stine.