UG Seminar - Lie Groups, Matt Burfitt (Southampton) Seminar

Event details

Abstract: 'It is impossible to have studied a mathematics degree without being exposed to some group theory. In particular group operations must be invertible and so groups are used in physics to model symmetries in a system. A differentiable manifold is a space that satisfies suitably nice conditions, that allow calculus to be done in a more general setting than R^n. This makes manifolds good objects for modelling the world. In particular manifolds satisfy the condition that locally they look like R^n, for some fixed n. Think of a sphere, which globally is round, however sitting on the surface of the earth, which roughly is spherical, the surface appears flat. A Lie group, named after mathematician Sophus Lie, is a set with the (compatible) structure of both a group and a manifold. Lie groups are particularly important as they are the natural objects to describe continuous symmetries. You are probably already familiar with some common examples of Lie groups: R^n under addition and O(n) orthogonal matric (isometries of R^n) under matrix multiplication are both Lie groups. Under suitable conditions Lie groups can be thought of as constructed from simpler components in a similar way to discrete groups in group theory. Such simple components are called simple Lie groups. Remarkably mathematics have since the 1890’s known of a classification, whose proof has been improved over time, in particular by Eugene Dynkin to whom the modern classification by Dynkin diagrams is due. The classification contains the four classical classes of simple Lie group and five exceptional cases, which were discovered as a consequence of the classification.'