Cartan-Killing Classification(s)

Many fundamental objects in algebra/combinatorics have a "Dynkin-Diagram"-type classification. I will focus on what these look like for semisimple Lie Algebras and finite Coxeter groups, sketching the ideas involved in proving the (purely combinatorial) classification of Dynkin diagrams. If time permits, I will mention other objects in which Dynkin-type classifications naturally arise.