One of the most beautiful applications of group theory - and one almost never mentioned in an introductory mathematics course on the subject - is in physics, where an elementary particle can be thought of as an irreducible representation of a group. Though a vast subject, I would like to give the flavor by focusing on a particular method of obtaining a Lie group from its associated Lie algebra, called the exponential mapping technique. I will use this method to derive the Lorentz boost matrices in special relativity, and time permitting may discuss a similar correspondence in quantum mechanics and quantum field theory.
This talk should be accessible to everyone - in particular no knowledge of physics is required.