We prove that the mirror map is the Strominger–Yau–Zaslow map for every toric Calabi–Yau surface. As a consequence, one obtains an enumerative meaning of the mirror map. This involves computing genus-0 open Gromov–Witten invariants, which is done by relating them with closed Gromov–Witten invariants via compactification and using an earlier computation by Bryan–Leung.

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In this paper we continue our study of hopficity begun in [1], [2], [3], [4] and [5]. Let$A$be hopfian and let$B$have a cyclic center of prime power order. We improve Theorem 4 of [2] by showing that if$B$has finitely many normal subgroups which form a chain (we say$B$is$n$-normal), then$AxB$is hopfian. We then consider the case when$B$is a$p$-group of nilpotency class 2 and show that in certain cases$AxB$is hopfian.

We analyze a family of singular Schrdinger operators with local singular interactions supported by a hypersurface n, n 2, being the boundary of a Lipschitz domain, bounded or unbounded, not necessarily connected. At each point of the interaction is characterized by four real parameters, the earlier studied case of - and -interactions being particular cases. We discuss spectral properties of these operators and derive operator inequalities between those referring to the same hypersurface but different couplings and describe their implications for spectral properties.

We review several recent results concerning two-dimensional systems which exhibit a transport induced by a one-dimensional perturbation of a homogeneous magnetic field. The first concerns the local Iwatsuka model, where a charged particle interacts with a field which is homogeneous outside a finite strip and translationally invariant along it: we present two new sufficient conditions for absolute continuity of the spectrum and show that in most cases the number of open spectral gaps is finite. In the second model the perturbation is a periodic array of point obstacles. In this case the Landau levels remain to be infinitely degenerate eigenvalues, and between them the system has bands of absolutely continuous spectrum.