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

#### Speaker:

Professor Brian Hall
#### Title:

A gentle introduction to symmetries and Lie groups
#### Abstract:

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