Classically geometers studied
orbit spaces of manifolds by group actions. If the action is “nice”,
then the quotient space is itself a manifold, and everyone is happy. But when
the action has fixed points the resulting quotient becomes singular. Often much
information about the group action is “lost in the singularity”.
Think of the silliest example: if somebody hands you a point and tells you that
that point is in fact the quotient of a point by the action of some group, you
would never be able to know what group it was just by looking at the resulting
quotient.

Orbifold geometry is a way to
encode some of the group theory in the geometry of the objects. One can think
of manifolds as geometric objects that are locally modelled as quotients of
manifolds by group actions, or as often is done these days, in categorical
terms as reprsenting some moduli functor.

In this class we’ll
start from the very basics of the theory, and hopefully end up discussing some
recent stuff by the end of the semester. And yes, incredibly we have a
textbook: