6 ECTS credits
150 h study time
Offer 1 with catalog number 4013293FNR for all students in the 2nd semester of odd academic years (e.g. 2013-2014) at a (F) Master - specialised level.
In this course we study several topics in algebra
that play an important role in recent research.
In order to do so we also have to deal with some
more classical topics that form the basis for
this. Some of these play a fundamental role in
algebraic geometry. The aim is to obtain a good
insight and intuition in the topics covered. It
is expected to obtain a solid understanding of
all the structures considered (including all
proofs). Furthermore, one is expected to develop
the skills to prove independently related
properties and structures.
In the first chapter we present several
constructions of associative rings, such as skew
(Laurent) polynomial rings, power series rings,
group- and semigroup rings, crossed products,
graded and filtered rings, quaternion algebras.
Also, several classes of groups will be
presented.
In the second chapter, we study the structure of
finite semigroups and linear semigroups.
Next we study discuss different open problems in
some of the mentioned classes of rings. We
describe the problem, give the necessary back
ground, prove some results and give an up to
date status of the problem.
Possible topics include: the Jacobson radical of
group and semigroup rings, crossed products and
graded rings (semisimplicity problem, Kothe and
Amitsur problems); when is a group algebra
Noetherian (with needed back ground on
polycyclic-by-finite groups); the zero divisor
problem of group algebras of torsion-free groups
(with necessary back ground on ordered groups and
uniques product groups, projective resolutions
and global dimension); prime and semiprime
group- and semigroup rings; Noetherian semigroep
algebras (with back ground on linear
semigroups); when are certain rings prime maximal
orders; set-theoretic solutions of the
Yang-Baxter equation; algebraic structure of
rings satisfying a polynomial identity; central
simple algebras.
1. Student knows and has insight into fundamental results in representation theory of algebras.
2. Student can look up related theory.
3. Student can analyse and understand related theory.
4. Student can make connections with other theories.
5. Student can synthesize and interpret results.
6. Student can independently consult and understand recent literature.
7. Student independently can prepare a mathematics text about another theory and report orally.
8. Student can analyze results.
9. Student independently can look up and solve exercises.
10. Student can think in function of problem solving.
The final grade is composed based on the following categories:
Other Exam determines 100% of the final mark.
Within the Other Exam category, the following assignments need to be completed:
Examination: oral exam and a project. A mark will only be assigned if the student has participated in all exams, tests and assignments.
The topic of the project will be discussed during the course. It can be a study of some topics related to non-commutative algebra or the study of a research article. The student should come up with a proposal after 4 weeks. After 8 weeks the student should present a written document showing sufficient progress. A final written document has to be submitted at the commencement of the oral exam. The exam starts with a short oral
presentation of the project. The student will be evaluated on the understanding of the material and on the broader view (insight) of the topic investigated.
This offer is part of the following study plans:
Master of Mathematics: Fundamental Mathematics (only offered in Dutch)