Abstract: The moduli of vector bundles on a curve has a beautiful structure called the Harder-Narasimhan stratification, which has many applications to studying the geometry of this classical moduli problem. The theory of Θ-stratifications provides an intrinsic description of this stratification that generalizes to other moduli problems. Recently, with Andres Fernandez-Herrero, we use this theory, along with a newly developed method of “infinite dimensional geometric invariant theory,” to generalize the Harder-Narasimhan stratification to the moduli of maps from a smooth curve to a quotient stack V/G, where G is a reductive group and V is an affine G-variety.

I will give an overview of these results. Then I will explain how to apply them in the case where V is a linear representation to give a formula for the K-theoretic gauged Gromov-Witten invariants of V in arbitrary genus. This includes some instances of quasi-maps invariants as a special case. If time allows, I will remark on the extension of these results to complete intersections.