Unipotent characters of finite groups

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My research is in the theory of groups. Groups are widely studied abstract mathematical objects. In some sense, group theory is the study of symmetry. Groups have applications in most areas of mathematics, as well as in physics, chemistry and other sciences. I will investigate certain aspects of representations of finite groups. A group may be seen as a collection of abstract elements that can potentially become symmetries. A representation is a way to associate symmetries of a concrete structure to those elements. Representation theory has proved to be an extremely valuable tool for investigating groups. Indeed, representations allowed mathematicians to find relatively easy proofs of deep theorems about groups. One of these theorems was proved by Georg Frobenius as early as 1901, and despite significant efforts, nobody has found a proof that does not use representations ever since. Representations are important in their own right and have numerous applications, for instance, in the study of molecular symmetry in chemistry.Two major types of finite group are of particular importance to this project: groups of Lie type and solvable groups. In the late 1970s Pierre Deligne and George Lusztig made ground-breaking discoveries in representation theory of groups of Lie type. They described representations using ideas from algebraic geometry, a very different branch of mathematics. On the other hand, representations of solvable groups have been successfully investigated by more direct methods.The underlying theme of this project is to bring these two approaches together. I will research representations of intermediate groups, each of which consists of two components, one solvable, and the other of Lie type. The key aim is to extend the geometric ideas of Deligne and Lusztig to a wider class of groups, thus making advances in representation theory of intermediate groups.