In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of graph Hamiltonians, being generic in a sense, inspires the question about the existence of graphs with a finite and nonzero number of spectral gaps. We show that the answer depends on the vertex couplings together with commensurability of the graph edges. A finite and nonzero number of gaps is excluded for graphs with scale invariant couplings; on the other hand, we demonstrate that graphs featuring a finite nonzero number of gaps do exist, illustrating the claim on the example of a rectangular lattice with a suitably tuned -coupling at the vertices.

We study the degeneration and the gluing of Kuranishi structures in Gromov-Witten theory under a symplectic cut. This leads us to a degeneration axiom and a gluing axiom for open Gromov-Witten invariants. They provide then a route to the construction of virtual fundamental chains via specialization. Comments on the equivalence of the degeneration formula of closed Gromov-Witten invariants by Li-Ruan/Li versus Ionel-Parker are given in the Appendix.

In this paper we construct a new time-periodic solution of the vacuum Einsteins field equations, this solution possesses physical singularities, i.e., the norm of the solutions Riemann curvature tensor takes the infinity at some points. We show that this solution is intrinsically time-periodic and describes a time-periodic universe with the time-periodic physical singularity. By calculating the Weyl scalars of this solution, we investigate new physical phenomena and analyze new singularities for this universal model.

We define a notion of stability for chiral ring of four dimensional N= 1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N= 1 field theory derived from D3 branes probing a three-fold singularity X, and show that the K stability which implies the existence of Ricci-flat conic metric on X is equivalent to the stability of chiral ring of the corresponding field theory.

We prove that for spacetimes solving the EinsteinMaxwell (EM) equations, the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In Nonlinear nature of gravitation and gravitational-wave experiments, Christodoulou showed that gravitational waves have a nonlinear memory. He discussed how this effect can be measured as a permanent displacement of test masses in a laser interferometer gravitational-wave detector. Christodoulou derived a precise formula for this permanent displacement in the Einstein vacuum (EV) case. We prove in Theorem 2.6 that for the EM equations this permanent displacement exhibits a term coming from the electromagnetic field. This term is at the same highest order as the purely gravitational term that governs the EV situation. On the other hand, in Section 3, we show that to leading order, the presence of the electromagnetic field