Cohomology of algebraic groups, Lie algebras, and finite groups of Lie type

Abstract: Let G be a reductive algebraic group over a field of prime characteristic. One can associate to G (or subgroups thereof) its Lie algebra, its Frobenius kernels, and the finite Chevalley group of points over a finite field. The representation theories of these structures are highly interconnected. This expository article will focus specifically on the cohomology theories of these structures and the relationships between them with the aim of highlighting a few key developments over the past 20 years and related open questions.