Reductive algebraic groups
The structure of reductive algebraic groups is an active research area with several open areas of enquiry. Open questions exist concerning their subgroup structure, representation theory, Lie algebra structure and geometry.
Finite groups of Lie type
Finite groups of Lie type arise as the finite analogues of reductive linear algebraic groups, and the theories of these objects are closely intertwined. Finite groups of Lie type give rise to most finite simple groups, in an appropriate sense, which makes them fundamental objects of study in group theory. These groups are therefore objects of intensive study, and there are many open problems to investigate.
There is a natural geometric structure on the collection of homomorphisms from a fixed finitely generated group into a fixed linear algebraic group. Studying this structure can provide insights into the structure of each group.