Quasimaps and mirror theorems for GIT quotients

Ionut Ciocan-Fontanine

(University of Minnesota)

Celebrated work of Givental proves genus zero mirror theorems for toric varieties and for complete intersections in them.
These theorems express certain generating series of Gromov-Witten invariants (known as the small J-functions) in terms of some explicit I-functions which he introduced.
In these lectures I will present an approach to computing the Gromov-Witten invariants of (many) GIT quotients via the theory of stable quasimaps (the toric case corresponds to quotients by abelian groups). I will explain how this theory leads to generalizations in many directions of Givental's results.