1 Lie Groups Definition and properties of Lie groups 2 Torus Actions Actions of a torus and weight space decomposition 3 Root Systems Root systems obtained from the weight decomposition of the adjoint representation 4 Borel Subgroups Dynkin diagrams, ADE classification, and flag varieties 5 Bruhat Decomposition Bruhat decomposition, homogeneous space cell decomposition, parabolic subgroups, and Grassmannian Schubert ... 6 Richardson Varieties and Peterson Varieties Richardson varieties and Peterson varieties: definitions, dimensions, intersection theory, and stratification