over a henselian discrete valuation ring, then the constant sheaf shifted by is an étale perverse sheaf. A simple example Let X be a disk around the origin in stratified so that the origin is the unique singular stratum. Then the category of perverse sheaves on X is equivalent to the category of diagrams of vector spaces where and are invertible. More generally, quivers can be used to describe perverse sheaves. Properties The category of perverse sheaves is an abelian subcategory of the (non-abelian) derived category of sheaves, equal to the core of a suitable t-structure, and is preserved by Verdier duality. The bounded derived category of perverse l-adic sheaves on a scheme X is equivalent to the derived category of constructible sheaves and similarly for sheaves on the complex analytic space associated to a scheme X/C. Applications Perverse sheaves are a fundamental tool for the geometry of singular spaces. Therefore, they are applied in a variety of mathematical areas. In the Riemann-Hilbert correspondence, perverse sheaves correspond to regular holonomic D-modules. This application establishes the notion of perverse sheaf as occurring 'in nature'. The decomposition theorem, a far-reaching extension of the hard Lefschetz theorem decomposition, requires the usage of perverse sheaves. Hodge modules are, roughly speaking, a Hodge-theoretic refinement of perverse sheaves. The geometric Satake equivalence identifies equivariant perverse sheaves on the affine Grassmannian with representations of the Langlands dual group of a reductive group G - see . A proof of the Weil conjectures using perverse sheaves is given in . String theory Massless fields in superstring compactifications have been identified with cohomology classes on the target space (i.e. four-dimensional Minkowski space with a six-dimensional Calabi-Yau (CY) manifold). The determination of the matter and interaction content requires a detailed analysis of the (co)homology of these spaces: nearly all massless fields in the effective physics model are represented by certain (co)homology elements. However, a troubling consequence occurs when the target space is singular. A singular target space means that only the CY manifold is singular as Minkowski space is smooth. Such a singular CY manifold is called a conifold as it is a CY manifold that admits conical singularities. Andrew Strominger observed (A. Strominger, 1995) that conifolds correspond to massless blackholes. Conifolds are important objects in string theory: Brian Greene explains the physics of conifolds in Chapter 13 of his book The Elegant Universe —including the fact that the space