65-401:  Advanced algebra - homological algebra and representation theory Lecturer: Christoph Schweigert Exercises: Aaron Hofer Max-Niklas Steffen Contents: 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 Aim: The goal of this lecture course is to present some algebraic tools used by many mathematicians. The two most important topics are: Representation theory: The idea 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/ss23/algebra2.html Prerequisites: 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 Exam: Individual oral exam in English or German, by individual appointment. Possible dates will be announced. Lecture notes: Lecture notes Problem sheets: will be provided in moodle. Please register in moodle. A password will be given in the first lecture. Time and Place: See information in Stine. Lectures start on Tuesday, April 4, 2023. Tutorials start on Monday, April 10 or on Tuesday, April 11, respectively. On Tuesday, June 6, our collaborative research group will be evaluated. There will be no lecture and no tutorials on this day.