Nicholas Proudfoot

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Preface The last decade has seen a number of exciting developments at the intersection of commutative algebra with combinatorics. New methods have evolved out of an influx of ideas from such diverse areas as polyhedral geometry, theoretical physics, representation theory, homological algebra, symplectic geometry, graph theory, integer programming, symbolic(More)
Given a matroid M represented by a linear subspace L ⊂ C n (equivalently by an arrangement of n hyperplanes in L), we define a graded ring R(L) which degenerates to the Stanley-Reisner ring of the broken circuit complex for any choice of ordering of the ground set. In particular, R(L) is Cohen-Macaulay, and may be used to compute the h-vector of the broken(More)
We demonstrate that distinct forms of the yeast chromatin-remodeling enzyme Isw1p sequentially regulate each stage of the transcription cycle. The Isw1a complex (Iswlp/Ioc3p) represses gene expression at initiation through specific positioning of a promoter proximal dinucleosome, whereas the Isw1b complex (Iswlp/Ioc2p/Ioc4p) acts within coding regions to(More)
We consider an orbifold X obtained by a Kähler reduction of C n , and we define its " hyperkähler analogue " M as a hyperkähler reduction of T * C n ∼ = H n by the same group. In the case where the group is abelian and X is a toric variety, M is a toric hyperkähler orbifold, as defined in [BD], and further studied in [K1, K2] and [HS]. The variety M carries(More)
We reexamine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical context is necessary, many familiar features survive. These include a(More)
We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkähler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to quiver varieties, and compute as an(More)
We define and study category O for a symplectic resolution, generalizing the classical BGG category O, which is associated with the Springer resolution. This includes the development of intrinsic properties paralleling the BGG case, such as a highest weight structure and analogues of twisting and shuffling functors, along with an extensive discussion of(More)
Given an n-tuple of positive real numbers (α 1 ,. .. , α n), Konno [K2] defines the hyperpolygon space X(α), a hyperkähler analogue of the Kähler variety M (α) parametrizing polygons in R 3 with edge lengths (α 1 ,. .. , α n). The polygon space M (α) can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension(More)