Graduate Student Seminar, 4:30 pm August 30, 2004; HH127


Professor Brian Hall


A gentle introduction to symmetries and Lie groups


Symmetry is everywhere in mathematics and physics. In some cases, the group of symmetries is finite; for example, the symmetry group of a regular dodecahedron has 120 elements. In many cases, however, the symmetry group is infinite, indeed, "continuous," meaning that is is described by several continuous real paramters. For example, translational symmetry is described by the group R^3, points of which are described by three real parameters. In mathematics, such "continuous" groups are called "Lie groups."

I will discuss various examples of Lie groups that arise as symmetry groups, including the translation group, the rotation group, the symmetry groups of the Euclidean and non-Euclidean planes, and the Lorentz group, which arises in special relativity. I will then give an introduction to the mathematical tools that one uses to analyze such groups, notably, the "Lie algebra." I will put things into the context of groups of matrices, where very little background is required to get started.

I will bring in various models which, besides being fun to play with, demonstrate some of the concepts involved.

To volunteer to give a talk, or for any other questions regarding this schedule, contact Sara Miller