- G-complete reducibility
This area of research investigates the relationship between subgroups of linear algebraic groups, representation theory and geometric invariant theory. To learn more, one possible starting point is this survey article.
Subgroup structure of reductive algebraic groups
This area forms part of an effort dating back to work of E.B. Dynkin in the 1950s, classifying various subgroups of reductive groups. Some of my contributions to this area are here (arXiv) and here (arXiv).
Finitely-generated groups and their images in algebraic groups
This area studies finitely-generated subgroups of linear algebraic groups. These can be studied directly via abstract group theory, algebraic geometry and geometric invariant theory. On the other hand, finitely-generated subgroups are images of finitely generated groups, and so this can also be viewed as a problem in geometric group theory. My introduction to this area was in my Masters thesis at Imperial College, which spawned this note.
This area produces and uses algorithms to solve problems in algebra, particularly group theory. My modest contributions include methods for enumerating feasible characters of finite subgroups on modules for exceptional algebraic groups (here), and methods for enumerating elementary abelian subgroups in simple algebraic groups (here).