I am currently doing a course on Lie groups, Lie Algebras and Representation theory based on Brian Hall's book of the same name. We should cover upto chapter 4/5 in this book by the end of the semester. Now for this course, our lecturer has suggested that we come up with a final project in the form of an approximately 15-page essay on any topic that we like related to Lie algebras. The difficulty of course is in choosing such a topic, perhaps those more experienced/familiar with the literature can help in suggesting one. So far, the following three suggestions have come up:
A final project related to the differential geometry side of things, i.e. matrix Lie groups as manifolds, flows, vector fields,etc.
A final project related to Algebraic Topology, e.g. perhaps classifying higher homotopy groups of the classical groups $(\textrm{SO}(n),\textrm{O}(n),\textrm{GL}_n, \textrm{Sp}_n$ etc).
A final project related to Algebraic Groups, suggestions for a final topic have been for example "What is a Reductive Group".
The list above is (possibly) non-exhaustive. As far as Algebraic groups go, I have had a look at the books by Humphreys, Borel and Tom Springer as well as the notes of James Milne. At this moment, Springer's book looks the most accessible with just 20 pages or so of algebraic geometry in the beginning.
My question is: What would be a good topic to look at combining Lie algebras and Algebraic Groups? Also can anyone suggest any good books/course notes/ material that I can look at apart from what I listed above?
Thanks.
Edit: I would add that this question may also be for suggestions on further topics in Lie Theory.