Nicholas Proudfoot

Moduli spaces for Bondal quivers

Given a sufficiently nice collection of sheaves on an algebraic variety V, Bondal explained how to build a quiver Q along with an ideal of relations in the path algebra of Q such that the derived category of representations of Q subject to these relations is equivalent to the derived category of coherent sheaves on V. We consider the case in which these sheaves are all locally free and study the moduli spaces of semistable representations of our quiver with relations for various stability conditions. We show that V can often be recovered as a connected component of such a moduli space and we describe the line bundle induced by a GIT construction of the moduli space in terms of the input data. In certain special cases, we interpret our results in the language of topological string theory.

Moduli spaces for D-branes at the tip of a cone

For physicists: We show that the quiver gauge theory derived from a Calabi-Yau cone via an exceptional collection of line bundles on the base has the original cone as a component of its classical moduli space. For mathematicians: We use data from the derived category of sheaves on a Fano surface to construct a quiver, and show that its moduli space of representations has a component which is isomorphic to the anticanonical cone over the surface.

Hyperpolygon spaces and their cores

Given an n-tuple of positive real numbers (α 1 ,.., an), Konno (2000) defines the hyperpolygon space X(a), a hyperkahler analogue of the Kahler variety M(a) parametrizing polygons in R 3 with edge lengths (α 1 ,..., an). The polygon space M(α) can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, X(a) is the hyperkahler quiver variety defined by Nakajima. A quiver variety admits a natural C*-action, and the union of the precompact orbits is called the core. We study the components of the core of X(α), interpreting each one as a moduli space of pairs of polygons in R 3 with certain properties. Konno gives a presentation of the cohomology ring of X(a); we extend this result by computing the C*-equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.

The hypertoric intersection cohomology ring

We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset.

The equivariant Kazhdan–Lusztig polynomial of a matroid

We define the equivariant Kazhdan-Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank.

Resolving Toric Varieties with Nash Blowups

It is a long-standing question whether an arbitrary variety is desingularized by finitely many normalized Nash blowups. We consider this question in the case of a toric variety. We interpret the normalized Nash blowup in polyhedral terms, show how continued fractions can be used to give an affirmative answer for a toric surface, and report on a computer investigation in which over a thousand 3- and 4-dimensional toric varieties were successfully resolved.

Poisson–de Rham homology of hypertoric varieties and nilpotent cones

We prove a conjecture of Etingof and the second author for hypertoric varieties that the Poisson–de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that this conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson–de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2-variable Poisson–de Rham–Poincaré polynomial and prove that it is equal to a specialization of an enrichment of the Tutte polynomial of a matroid that was introduced by Denham (J Algebra 242(1):160–175, 2001). We also compute this polynomial for S3-varieties of type A in terms of Kostka polynomials, modulo a previous conjecture of the first author, and we give a conjectural answer for nilpotent cones in arbitrary type, which we prove in rank less than or equal to 2.

Hypertoric Poisson homology in degree zero

Etingof and Schedler formulated a conjecture about the degree zero Poisson homology of an affine cone that admits a projective symplectic resolution. We strengthen this conjecture in general and prove the strengthened version for hypertoric varieties. We also formulate an analogous conjecture for the degree zero Hochschild homology of a quantization of such a variety.

ALL THE GIT QUOTIENTS AT ONCE

Let G be an algebraic torus acting on a smooth variety V. We study the relationship between the various GIT quotients of V and the symplectic quotient of the cotangent bundle of V.

The Orlik-Terao Algebra and the Cohomology of Configuration Space

ABSTRACT We give a recursive algorithm for computing the Orlik-Terao algebra of the Coxeter arrangement of type An − 1 as a graded representation of Sn, and we give a conjectural description of this representation in terms of the cohomology of the configuration space of n points in SU(2) modulo translation. We also give a version of this conjecture for more general graphical arrangements.