Symmetry is a central theme of modern mathematics and theoretical physics. The intuitive idea of symmetry has been abstracted by mathematicians in to a more powerful, general concept – group theory – by means of which we can analyze symmetries, not only of faces and snowflakes, but also of equations or mathematical structures. Mathematicians before the nineteenth century had succeeded in extending the solution of quadratic equations, that you learn in high school, to cover equations with third or fourth powers of the unknown; but the solution of polynomial equations with fifth and higher powers had eluded everyone for hundreds of years. In 1832, Galois showed that the ability to find a solution of an equation was intimately tied to the symmetries of that equation. Ever since, symmetry has played a unifying and illuminating role, up to the present day when it is central to much of mathematics and to our fundamental theories of physics (for example, underpinning the “Eightfold Way”, which is our periodic table for subatomic particles).
This class will cover: the basics of finite group theory, group actions and representations; techniques of generators and relations; the analysis of tiling patterns and crystal structures; the beginnings of galois theory; and the beginnings of continuous symmetries and their application to physics. The topics will be similar to a standard first course in abstract algebra, but with a difference in emphasis.