On the symplectic eightfold associated to a Pfaffian cubic fourfold

We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic Pfaffian cubic Y a birational map Z ---> Hilb^4(X), where X is the K3 surface associated to Y by Beauville and Donagi. We interpret Z as a moduli space of complexes on X and observe that at some point of Z, hence on a Zariski open subset, the complex is just the ideal sheaf of four points.Comment: 9 pages. Minor changes; to appear in Crelle as an appendix to 1305.017

Vertex Operators, Grassmannians, and Hilbert Schemes

We describe a well-known collection of vertex operators on the infinite wedge representation as a limit of geometric correspondences on the equivariant cohomology groups of a finite-dimensional approximation of the Sato grassmannian, by cutoffs in high and low degrees. We prove that locality, the boson-fermion correspondence, and intertwining relations with the Virasoro algebra are limits of the localization expression for the composition of these operators. We then show that these operators are, almost by definition, the Hilbert scheme vertex operators defined by Okounkov and the author in \cite{CO} when the surface is $\mathbb{C}^2$ with the torus action $z\cdot (x,y) = (zx,z^{-1}y)$.Comment: 20 pages, 0 figure

Nonprofit Performance: Accounting for the Agency of Clients

Performance is a key concern for nonprofits providing human services. Yet our understanding of what drives performance remains incomplete. Existing outcome measurement systems track the programmatic activities staff complete and the extent to which participants respond in programmatically intended ways. However, clients do not just receive services and respond as intended and staff do not simply complete program activities. Drawing on a data set of 47 interviews with frontline staff in eight human service nonprofits, we show how frontline staff work in a partnership with clients to set an agenda for change and achieve desired results. We call this co-determination work and argue that it represents a critical and often neglected dimension of nonprofit performance

Second quantized Frobenius algebras

We show that given a Frobenius algebra there is a unique notion of its second quantization, which is the sum over all symmetric group quotients of n--th tensor powers, where the quotients are given by symmetric group twisted Frobenius algebras. To this end, we consider the setting of Frobenius algebras given by functors from geometric categories whose objects are endowed with geometric group actions and prove structural results, which in turn yield a constructive realization in the case of n--th tensor powers and the natural permutation action. We also show that naturally graded symmetric group twisted Frobenius algebras have a unique algebra structure already determined by their underlying additive data together with a choice of super--grading. Furthermore we discuss several notions of discrete torsion andshow that indeed a non--trivial discrete torsion leads to a non--trivial super structure on the second quantization.Comment: 39p. Latex. New version fixes sign mistake and includes the full description of discrete torsio

Self-sorting of two imine-based metal complexes: Balancing kinetics and thermodynamics in constitutional dynamic networks

A major hurdle in the development of complex constitutional dynamic networks (CDNs) is the lack of strategies to simultaneously control the output of two (or more) interconnected dynamic processes over several species, namely reversible covalent imine bond formation and dynamic metal–ligand coordination. We have studied in detail the self-sorting process of 11 constitutional dynamic libraries containing two different amines, aldehydes and metal salts into two imine-based metal complexes, having no overlap in terms of their compositions. This study allowed us to determine the factors influencing the fidelity of this process (concentration, electronic and steric parameters of the organic components, and nature of the metal cations). In all 11 systems, the outcome of the process was primarily determined by the ability of the octahedral metal ion to select its pair of components from the initial pool of components, with the composition of the weaker tetrahedral complex being imposed by the components rejected by the octahedral metal ions. Different octahedral metal ions required different levels of precision in the “assembling instructions” provided by the organic components of the CDN to guide it towards a sorted output. The concentration of the reaction mixture, and the electronic and steric properties of the initial components of the library were all found to influence the lifetime of unwanted metastable intermediates formed during the assembling of the two complexes