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My primary research is in finite group theory and the representation theory of finite groups. Group theory can be viewed as the study of symmetries of an object, such as those coming from nature, art, communication networks, or any other place that symmetry might play a role. Representation theory is a tool used to better understand the structure of a group and the symmetries it represents by giving us a way to view, in some sense, an abstract group as a group of matrices whose structure is often easier to understand. In particular, the irreducible representations form the “building blocks” of all representations. (Click here for a short introduction to the area by my colleagues Eugenio Giannelli and Jay Taylor.)
Most of my research involves the irreducible representations of finite groups of Lie type, sometimes called finite reductive groups. These groups are analogues of Lie groups over finite fields and form the largest collection of finite simple groups. I have been particularly interested in problems concerning local-global conjectures in character theory and problems involving the action of Galois automorphisms of characters.
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