Scott Warrander

Project Title: Kac-Moody Groups and Geometric Representation Theory

A fundamental aspect of modern mathematics is studying symmetries: transformations of an object which preserve some notion of an underlying structure. Smooth, continuous symmetry transformations are of particular interest, both in pure mathematics and in more applied fields. We formalise the notion of symmetry using an algebraic structure called a group, and in the case of smooth symmetry transformations, a Lie group. Finite dimensional Lie groups are very well understood at this point, and so my project will focus on a large class of infinite dimensional Lie groups called Kac-Moody groups.

In practice, understanding Lie groups usually means understanding their representation theory, which is the theory of how a group can act on vector spaces by linear transformations. Representation theory allows us to apply techniques of linear algebra – a straightforward and well understood field compared to group theory – to answer questions about groups. Something of a mystery in modern mathematics is the phenomenon of Langlands duality, which forms a bridge between representation theory and algebraic geometry and topology. Essentially, problems of representation theory can be tackled by passing to a seemingly unrelated dual situation (and vice versa), giving us a very powerful and not yet fully understood tool for solving problems in both areas. This project will be about exploring Langlands duality for the case of Kac-Moody groups.