Student Algebra and Representation Theory Seminar (StARTS)

The objective is to give a good seminar lecture, while giving as many meaningful examples (and nonexamples) of reductive groups as possible.

The speaker has fifty-seven minutes to give a lecture including examples of linear algebraic groups and prove which properties they have. Examples are awarded points subject to the following rules:

1. A given example of a linear algebraic group, including a faithful representation, is awarded one point when it is first rigorously shown to be reductive or not reductive.
2. The speaker can assume only that the listeners know the definition of a linear algebraic group and all of its contingent definitions in Waterhouse and contingent foundations within set theory and category theory. In particular the speaker must define what a reductive group is to win any points, and can not use the notions of unipotent radicals, of root data, or of Borel subgroups, without defining them.
3. If an example of a linear algebraic group G descends to a group scheme G_R over a commutative ring R, then no base change of G_R, other than G, may be counted as an example.
4. No more than one nontrivial unipotent group may be counted as an example of a linear algebraic group.
5. The speaker can state and prove as many theorems, propositions, lemmas, etc. as time permits. BUT, any theorem, proposition, lemma, etc. may only be invoked (explicitly or implicitly) once to prove that some example is or is not reductive.
6. All examples of linear algebraic groups must be connected and defined over an algebraically closed field to be awarded any points. The speaker does not need to show that an example is connected. The field of definition can change between examples.
7. Assuming the speaker has given an example a linear algebraic group G, and has shown that G is reductive, additional points can be awarded to G as follows:
   1. If the speaker has defined the notion of Borel subgroups of algebraic groups, then G is to be awarded one additional point for giving an example of a subgroup B of G and showing that B is a Borel subgroup of G.
   2. If the speaker has defined the notion of Borel subgroups and unipotent radicals of algebraic groups, and a Borel subgroup B of G is given, then G is to be awarded one additional point for giving an example of a subgroup U of B and showing that U is the unipotent radical of B.
   3. If the speaker has defined the notion of root data, then G is to be awarded two additional points if the root datum of G is given explicitly by the speaker.
8. No points are to be awarded based on any mathematical expression given in the form of priorly-prepared visual aides (digital or physical slides).

The speaker must be a person, and must not make use of artificial intelligience technology outside of medical necessity, in order to win any points.

The speaker wins the sum of all points awarded to any example of a connected linear algebraic group given in the lecture.

All rules above are to be interpreted at the collective discretion of the listeners.

If the speaker wins more than twelve points or achieves a high-score, their name is to be immortalized in the HALL OF FAME, and festivities are to commence.