Groups of continuous transformations, now known as Lie groups, have their origins in the work of Sophus Lie in the late 19th century, partly motivated by the wish to develop Galois theory for differential equations. Since then, the study of Lie groups and Lie algebras has been a central theme in mathematics. In the 1950s, the ``analytic theory’’ was extended so that it also makes sense over arbitrary algebraically closed fields and spawned the area of mathematics now known as algebraic Lie theory. Nowadays, algebraic Lie theory encompasses a variety of different areas; for example, it includes the structure theory and representation theory of algebraic groups, finite groups of Lie type, Lie algebras, and many related algebras. A common theme in the area is a strong interplay between algebra, geometry and combinatorics.
Throughout his career, Simon Goodwin has developed interests in a range of topics in Lie theory and representation theory. His research is centered around questions in the representation theory of Lie algebras and algebraic groups. Recently he has been interested in the representation theory of modular Lie algebras and of Lie superalgebras, and in particular the application of the theory of W-algebras to these areas. This research has involved a blend of algebraic, combinatorial and geometric methods.
When studying the representation theory of semisimple algebraic groups and Lie algebras over fields of positive characteristic, one difference from the theory in characteristic zero is that there now exist finite-dimensional simple modules for the Lie algebra which aren't seen by the algebraic group. We call these representations non-restricted. Matthew Westaway’s research has largely focused on deepening our understanding of these simple modules. Among his contributions, he has developed a theory of non-restricted representations which is compatible with higher Frobenius kernels (mirroring the known compatibility with the first Frobenius kernel) and has generalised some key aspects of the theory of restricted representations to the non-restricted world. Since the representation theory of finite W-algebras provides a characteristic zero analogue of the non-restricted representation theory of Lie algebras, he also works in this area, in which he collaborates with Simon Goodwin and Lewis Topley (Bath).