We propose a number of techniques for obtaining a global ranking from data that may be incomplete and imbalanced — characteristics that are almost universal to modern datasets coming from e-commerce and internet applications. We are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data. From raw ranking data, we construct...

Assuming suitable convergence properties for the Gromov-Witten potential of a Calabi-Yau manifold X, one may construct a polarized variation of Hodge structure over the complexified Kähler cone of X. In this paper we show that, in the case of fourfolds, there is a correspondence between “quantum potentials” and polarized variations of Hodge structures that degenerate to a maximally unipotent bo...

In [4], Lin and Sjamaar show how to use symplectic Hodge theory to obtain canonical equivariant extensions of closed forms in Hamiltonian actions of compact connected Lie groups on closed symplectic manifolds which have the strong Lefschetz property. In this paper, we show how to do the same using classical Hodge theory. This has the advantage of applying far more generally. Our method makes us...

This paper constructs a Hodge theory of noncompact topologically tame manifolds M . The main result is an isomorphism between the de Rham cohomology with compact supports of M and the kernel of the Hodge–Witten–Bismut Laplacian △μ associated to a measure dμ which has sufficiently rapid growth at infinity on M . This follows from the construction of a space of forms associated to △μ which satisf...

Let (M, I,ω,Ω) be a nearly Kähler 6-manifold, that is, an SU(3)-manifold with the (3,0)-form Ω and the Hermitian form ω which satisfies dω = 3λReΩ, d ImΩ = −2λω, for a non-zero real constant λ. We develop an analogue of Kähler relations on M , proving several useful identities for various intrinsic Laplacians onM . WhenM is compact, these identities bring powerful results about cohomology of M ...

The antique origins of Algebraic Geometry lie in the study of solution sets of polynomial equations, in which complex, symplectic, and arithmetic geometry are bound tightly together. Many of the most spectacular recent developments in the subject have occurred through the consideration of these aspects in tandem: for example, the duality between symplectic and complex geometry that is mirror sy...

We prove that when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of harmonic one-forms. For example, this is the case for complete Kähler manifolds for which the symplectic form has an appropriate decay at infinity. This extends ...