The Witten equation, mirror symmetry and quantum singularity theory

For any non-degenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity A_{r-1}. We also resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual; and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.Comment: To appear in Annals of Mathematics. Includes resolution of the Witten ADE integrable hierarchies conjecture and Witten's ADE self-mirror conjecture. Several corrections and clarification

Agegraphic Dark Energy Model in Non-Flat Universe: Statefinder Diagnostic and w−w′w-w^{\prime} Analysis

We study the interacting agegraphic dark energy (ADE) model in non-flat universe by means of statefinder diagnostic and $w-w^{\prime}$ analysis. First, the evolution of EoS parameter ($w_d$) and deceleration parameter ($q$) in terms of scale factor for interacting ADE model in non-flat universe are calculated. Dependence of $w_d$ on the ADE model parameters $n$ and $\alpha$ in different spatial curvatures is investigated. We show that the evolution of $q$ is dependent on the type of spatial curvature, beside of dependence on parameters $n$ and $\alpha$. The accelerated expansion takes place sooner in open universe and later in closed universe compare with flat universe. Then, we plot the evolutionary trajectories of the interacting ADE model for different values of the parameters $n$ and $\alpha$ as well as for different contributions of spatial curvature, in the statefinder parameters plane. In addition to statefinder, we also investigate the ADE model in non-flat universe with $w-w^{\prime}$ analysis.Comment: 20 pages, 4 figures, 2 tables, International Journal of Modern Physics D accepte