Math 676: Orbifolds

WHEN: MWF 11 – 11:50 pm WHERE: Weber 223 // in the oval with good weather WHAT: Some good fun with orbifolds

Orbi-donut

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:

HOMEWORK (really? Really!)

Hwk1

VARIOUS REFERENCES FOR THIS CLASS:

1. Orbifolds, Sheaves and Groupoids (Moerdijk and Pronk)

2. Orbifold Cohomology of ADE Singularities (Fabio Perroni’s PhD thesis)

3. Orbifold Cohomology for Global Quotients (Fantechi and Goettsche)