Hunter Dinkins (MIT): Enumerative 3d mirror symmetry of bow varieties

The perspective of enumerative geometry on 3d mirror symmetry (or symplectic duality) predicts an "equivalence" between curve counts in a Higgs branch and those in the corresponding Coulomb branch. In our current understanding, such an equivalence requires the existence of symplectic resolutions of both branchs. In these cases, the enumerative duality provides a very strong refinement of other aspects of 3d mirror symmetry (e.g. bijections between torus fixed points, the Hikita conjecture and its quantum version). 

In this talk, I will give an overview of the enumerative perspective on 3d mirror symmetry and discuss my recent joint work with Tommaso Botta, in which we prove the duality for finite type A bow varieties. Bow varieties provide the largest currently known setting where the appropriate curve counts can be defined and their equivalence precisely formulated. Our proof combines geometric, combinatorial, and analytic arguments to eventually reduce to the case of the cotangent bundle of the complete flag variety. Time permitting, I will also discuss ongoing work to incorporate "descendant insertions" into the statements by using Hecke modifications of vector bundles.