Dwork families and D\mathcal{D}-modules

A Dwork family is a one-parameter monomial deformation of a Fermat hypersurface. In this paper we compute algebraically the invariant part of its Gauss-Manin cohomology under the action of certain subgroup of automorphisms. To achieve that goal we use the algebraic theory of $\mathcal{D}$-modules, especially one-dimensional hypergeometric ones.Comment: 24 pages, final versio

Irregular Hodge filtration of some confluent hypergeometric systems

We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric $\mathcal{D}$-modules. We obtain in particular a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss-Manin systems of Laurent polynomials via Fourier-Laplace and Radon transformations.Comment: 32 pages, final versio

Lagrangian 3-torus fibrations

We prove that Mark Gross' topological Calabi-Yau compactifications can be made into symplectic compactifications. To prove this we develop a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. This produces pairs of compact symplectic 6-manifolds homeomorphic to mirror pairs of Calabi-Yau 3-folds together with Lagrangian fibrations whose underlying integral affine structures are dual

Crossed extensions of the corepresentation category of finite supergroup algebras

We present explicit examples finite tensor categories that are C_2-graded extensions of the corepresentation category of certain finite-dimensional non-semisimple Hopf algebras.Comment: 26 pages. An error in the product given in definition 4.7 was correcte

Variational approach to second-order impulsive dynamic equations on time scales

The aim of this paper is to employ variational techniques and critical point theory to prove some conditions for the existence of solutions to nonlinear impulsive dynamic equation with homogeneous Dirichlet boundary conditions. Also we will be interested in the solutions of the impulsive nonlinear problem with linear derivative dependence satisfying an impulsive condition.Comment: 17 page