This course gives an introduction to the classical matrix groups:
the orthogonal and special orthogonal groups, the unitary and
special unitary groups, and the symplectic group. The classical
groups are used as a vehicle to introduce and connect a wide variety
of topics within mathematics, ranging from linear and abstract
algebra to topology and geometry to representation theory. Topics
include the topology of the groups, Lie algebra structure,
Riemannian structure, the Lie bracket, and the adjoint
representation.
Prerequisites:
MATH 307 is required. MATH 308 would be helpful but is not
required; I will fill in group theory background as needed,
depending on the audience.
Course plan:
Mondays and Wednesdays will be lectures. Fridays will be student
presentations of problems. You should do as many of the problems
from the book as you can as we go along (I'll let you know which
ones we've covered enough for as we go), and on Fridays we will go
through the problems with students volunteering to present their
solutions. I will not assign specific problems to specific
students, but you are expected to present on a regular basis.
Attendance:
Attendance at every class is required.
If you ever need to miss class because of travel or other
pre-arranged circumstances, let me know as soon as you know
yourself. If you have to miss class due to illness or other
emergency, let me know as soon as you are able.
Reading:
I won't be able to cover every detail from
the book in class; you are expected to read the book as we go.
Scribing:
Each Friday, one student will be the designated scribe. It
will be their job to take good notes on the solutions presented and
to type up clean versions of all those solutions. The scribe will
submit their typed solutions to me one week after the problem
session at which they served as scribe. I will give notes and
suggested revisions, and the final version will be submitted to me
and to the rest of the class one week later.