Variations of Hodge Structure Considered as an Exterior Differential System: Old and New Results

This paper is a survey of the subject of variations of Hodge structure (VHS) considered as exterior differential systems (EDS). We review developments over the last twenty-six years, with an emphasis on some key examples. In the penultimate section we present some new results on the characteristic cohomology of a homogeneous Pfaffian system. In the last section we discuss how the integrability conditions of an EDS affect the expected dimension of an integral submanifold. The paper ends with some speculation on EDS and Hodge conjecture for Calabi-Yau manifolds

Extremal degenerations of polarized Hodge structures

We describe a Hodge theoretic approach to the question: In what ways can a smooth projective variety degenerate?Comment: v.2 (typos corrected), comments welcom

A Novel 2.5D Culture Platform to Investigate the Role of Stiffness Gradients on Adhesion-Independent Cell Migration

Current studies investigating the role of biophysical cues on cell migration focus on the use of culture platforms with static material parameters. However, migrating cells in vivo often encounter spatial variations in extracellular matrix stiffness. To better understand the effects of stiffness gradients on cell migration, we developed a 2.5D cell culture platform where cells are sandwiched between stiff tissue culture plastic and soft alginate hydrogel. Under these conditions, we observed migration of cells from the underlying stiff substrate into the alginate matrix. Observation of migration into alginate in the presence of integrin inhibition as well as qualitative microscopic analyses suggested an adhesion-independent cell migration mode. Observed migration was dependent on alginate matrix stiffness and the RhoA-ROCK-myosin-II pathway; inhibitors specifically targeting ROCK and myosin-II arrested cell migration. Collectively, these results demonstrate the utility of the 2.5D culture platform to advance our understanding of the effects of stiffness gradients and mechanotransductive signaling on adhesion-independent cell migration

Period Mappings and Ampleness of the Hodge line bundle

We discuss progress towards a conjectural Hodge theoretic completion of a period map. The completion is defined, and we conjecture that it admits the structure of a compact complex analytic variety. The conjecture is proved when the image of the period map has dimension 1,2. Assuming the conjecture holds, we then prove that the augmented Hodge line bundle extends to an ample line bundle on the completion. In particular, the completion is a projective algebraic variety that compactifies the image, analogous to the Satake-Baily-Borel compactification.Comment: 62 pages. v2 significant revision of the initial submission (v1); v3 further improvements and new references adde

Matrix elements and duality for type 2 unitary representations of the Lie superalgebra gl(m|n)

The characteristic identity formalism discussed in our recent articles is further utilized to derive matrix elements of type 2 unitary irreducible $gl(m|n)$ modules. In particular, we give matrix element formulae for all gl(m|n) generators, including the non-elementary generators, together with their phases on finite dimensional type 2 unitary irreducible representations. Remarkably, we find that the type 2 unitary matrix element equations coincide with the type 1 unitary matrix element equations for non-vanishing matrix elements up to a phase.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1311.424

Reduced Wigner coefficients for Lie superalgebra gl(m|n) corresponding to unitary representations and beyond

In this paper fundamental Wigner coefficients are determined algebraically by considering the eigenvalues of certain generalized Casimir invariants. Here this method is applied in the context of both type 1 and type 2 unitary representations of the Lie superalgebra gl(mjn). Extensions to the non-unitary case are investigated. A symmetry relation between two classes of Wigner coefficients is given in terms of a ratio of dimensions.Comment: 17 page