Fano polytopes are the convex-geometric objects corresponding to toric Fano varieties. We give a brief survey of classification results for di↵erent classes of Fano polytopes.

Abstract Recently it has been observed that the group E7 can be used to describe a special type of quantum entanglement of seven qubits partitioned into seven tripartite systems. Here we show that this curious type of entanglement is entirely encoded into the discrete geometry of the Fano plane. We explicitly work out the details concerning a qubit interpretation of the E7 generators as represe...

We develop an idea of a local 3-edge-coloring of a cubic graph, a generalization of the usual 3-edge-coloring. We allow for an unlimited number of colors but require that the colors of two edges meeting at a vertex always determine the same third color. Local 3-edge-colorings are described in terms of colorings by points of a partial Steiner triple system such that the colors meeting at each ve...

Let H be a 3-uniform hypergraph of order n and size m, and let T be a subset of vertices of H. The set T is a strong transversal in H if T contains at least two vertices from every edge of H. The strong transversal number τs(H) of H is the minimum size of a strong transversal in H. We show that 7τs(H) ≤ 4n+ 2m, and we characterize the hypergraphs that achieve equality in this bound. In particul...

We review some recently established connections between the mathematics of black hole entropy in string theory and that of multipartite entanglement in quantum information theory. In the case of N = 2 black holes and the entanglement of three qubits, the quartic [SL(2)] 3 invariant, Cayley's hyperdeterminant, provides both the black hole entropy and the measure of tripartite entanglement. In th...

Understanding vanishing transmission in Fano resonances quantum systems and metamaterials, ultralow disordered media is key to elucidating optical interactions. Using analytic theory numerical simulations, we discover the topological structure motion under deformation of zeros complex energy plane. We demonstrate can be controlled create ultranarrow Lorentzian lines spectra time diverging rates...

In this Note, we announce a factorization result for equivariant birational morphisms between toric 4-folds whose source is Fano: such a morphism is always a composite of blow-ups along smooth invariant centers. Moreover, we show with a counterexample that, differently from the 3-dimensional case, even if both source and target are Fano, the intermediate varieties can not be chosen Fano. Sur la...

We study the properties of spatially localized and time-periodic excitations--discrete breathers--in Fermi-Pasta-Ulam (FPU) chains. We provide a detailed analysis of their spatial profiles and stability properties. We especially demonstrate that the Page mode is linearly stable for symmetric FPU potentials. A resonant interaction between a localized and delocalized perturbations causes weak but...

We study a waveguide array with an embedded nonlinear saturable impurity. We solve the impurity problem in closed form and find the nonlinear localized modes. Next, we consider the scattering of a small-amplitude plane wave by a nonlinear impurity mode, and discover regions in parameter space where transmission is fully suppressed. We relate these findings with Fano resonances and propose this ...