September 28  Lie groups,
the Lie algebra of a Lie group.
October 3  The Lie
algebra of a Lie group. An Example.
October 5  Lie group
Homomorphisms. Smooth Maps.
October X  Partitions of
Unity. Inverse Function Theorem.
November Y  Submanifolds,
Slices, and the Implicit Function Theorem.
Ordinary Differential
Equations  General existence and uniqueness theorem for
arbitrary, time-dependent systems of ordinary differential
equations.
November 16  Integral
Curves and Flows of Smooth Vector Fields.
November 21  Complete
Vector Fields/Group Actions/Dynamical Systems.
November 28 
Left-invariant Vector Fields and One Parameter Subgroups of Lie
Groups.
December 14  The
Exponential Map for General Linear Groups, Left-invariant Vector
Fields and One Parameter Subgroups of the general Linear
Group.
January U  Unitary and
Orthogonal groups: SU(2) (latitudes, longitudes, conjugacy classes,
maximal tori), the adjoint representation of SU(2), the
Hopf fibration.
February 28  Tensor
products, intertwining operators, and representations; Characters;
Examples -- the symmetric group on three letters, the unit circle
and Fourier seres.