Good day all,
I have to submit a report for a course in 'Applied Algebraic Structures' this period. The lecturer is a very good mathematician but his presentation style is very example-centric and I find myself without any real high-level understanding of the topics we've covered. This is making choosing a project topic difficult.
The idea of the project is to review a particular section of the course in greater detail than it is presented and to give a short seminar (10-15 minutes) on the topic.
The course syllabus is included at the bottom of the post (coarse overview, I've shorted this for the reader's benefit).
My bachelor's is in mechanical engineering but I'm currently doing a masters in applied mathematics. I want to try and focus on a subject area that is applicable to control theory in particular. A bit of googling shows that people have done some work on polynomial rings and Ore extensions (of polynomial rings?), so the idea doesn't seem daft at the surface level at least.
However, I'm having trouble outlining what I'd actually put in the project because I'm unable to string the topics of the course together coherently at the moment.
The definition of polynomial rings is very complex. Unless I misunderstand you can just take the set of all polynomials with complex coefficients and with normal addition and multiplication you have a commutative ring. I'm not sure when properties I could analyse or derive that would be relevant.
Ore extensions are not covered in the course, so I don't know much about them and I can't see from their definition what properties they'd have that make them useful in control theory.
I'd appreciate any ideas about what I could look into in this direction, or suggestions for other possibilities.
If the entire idea is not well founded I'm planning to just do the project on the first block of work on Lie Algebras since improving my understanding of the fundamentals seems like a good idea as well.
Course Content
Lie algebras, Lie groups and Lie symmetries
Non-commutative Analysis and Symmetry
Lie algebras and their deformations and generalizations
Reordering formulae and q-combinatorics
Quantum Planes
Schrödinger operators (canonical)
Schrödinger pairs or couples
Unitary Groups
von Neumann Theorem
Matrix Differential Equations and Dynamical Matrix Equations