Foam-like structure of the Universe

On the quantum stage spacetime had the foam-like structure. When the Universe cools, the foam structure tempers and does not disappear. We show that effects caused by the foamed structure mimic very well the observed Dark Matter phenomena. Moreover, we show that in a foamed space photons undergo a chaotic scattering and together with every discrete source of radiation we should observe a diffuse halo. We show that the distribution of the diffuse halo of radiation around a point-like source repeats exactly the distribution of dark matter around the same source, i.e. the DM halos are sources of the diffuse radiation

Density perturbations in the gas of wormholes

The observed dark matter phenomenon is attributed to the presence of a gas of wormholes. We show that due to topological polarization effects the background density of baryons generates non-vanishing values for wormhole rest masses. We infer basic formulas for the scattering section between baryons and wormholes and equations of motion. Such equations are then used for the kinetic and hydrodynamic description of the gas of wormholes. In the Newtonian approximation we consider the behavior of density perturbations and show that at very large distances wormholes behave exactly like heavy non-baryon particles, thus reproducing all features of CDM models. At smaller scales (at galaxies) wormholes strongly interact with baryons and cure the problem of cusps. We also show that collisions of wormholes and baryons lead to some additional damping of the Jeans instability in baryons

On Scattering of Electromagnetic Waves by a Wormhole

We consider scattering of a plane electromagnetic wave by a wormhole. It is found that the scattered wave is partially depolarized and has a specific interference picture depending on parameters of the wormhole and the distance to the observer. It is proposed that such features can be important in the direct search of wormholes

Skew Divided Difference Operators and Schubert Polynomials

We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with positive integer coefficients.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA