Z/2 harmonic 1-forms are generalizations of harmonic 1-forms that allow topological twisting around a subspace of codimension 2. These objects were introduced by Taubes to compactify the moduli spaces of solutions to generalized Seiberg-Witten equations, and they show up in many other gauge theoretical problems.

Donaldson showed there is a deformation theory for so-called non-degenerate Z/2-harmonic 1-forms. In this presentation we study the deformations of the remaining degenerate solutions. For a natural class of degenerate examples, we prove that after a suitable perturbation of the ambient Riemannian metric, the form can be deformed to a nearby non-degenerate Z/2-harmonic 1-form.

Hypertoric varieties are conical symplectic singularities equipped with a hamiltonian action of a torus of maximal possible dimension. They behave analogously to toric varieties in many ways, with the hamiltonian torus replacing the dense torus. Examples include: (crepant partial resolutions of) rational surface singularities of type A, the cotangent bundle of projective space, and Nakajima quiver varieties with the `all 1s' dimension vector.

Arbo and Proudfoot conjectured a combinatorial classification of hypertoric varieties by zonotopal tilings. In this talk I will discuss a proof of the conjecture.