A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold

We prove a generalized Bogomolov-Gieseker inequality as conjectured by Bayer, Macr\`i and Toda for the smooth quadric threefold. This implies the existence of a family of Bridgeland stability conditions.Comment: v1: 10 pages. v2: Updated references. v3: 11 pages. Several minor corrections. Accepted for publication in "Bulletin of the London Mathematical Society

Plasmonic Metamaterials: Physical Background and Some Technological Applications

New technological frontiers appear every year, and few are as intriguing as the field of plasmonic metamaterials (PMMs). These uniquely designed materials use coherent electron oscillations to accomplish an astonishing array of tasks, and they present diverse opportunities in many scientific fields. This paper consists of an explanation of the scientific background of PMMs and some technological applications of these fascinating materials. The physics section addresses the foundational concepts necessary to understand the operation of PMMs, while the technology section addresses various applications, like precise biological and chemical sensors, cloaking devices for several frequency ranges, nanoscale photovoltaics, experimental optical computing components, and superlenses that can surpass the diffraction limit of conventional optics

Characterizing Round Spheres Using Half-Geodesics

A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round

Three-manifolds with many flat planes

We discuss the rigidity (or lack thereof) imposed by different notions of having an abundance of zero curvature planes on a complete Riemannian 3-manifold. We prove a rank rigidity theorem for complete 3-manifolds, showing that having higher rank is equivalent to having reducible universal covering. We also study 3-manifolds such that every tangent vector is contained in a flat plane, including examples with irreducible universal covering, and discuss the effect of finite volume and real-analiticity assumptions.Comment: LaTeX2e, 24 pages, 7 figures, revised version. To appear in Trans. Amer. Math. So

Positively curved manifolds with large spherical rank

Rigidity results are obtained for Riemannian $d$-manifolds with $\sec \geqslant 1$ and spherical rank at least $d-2>0$. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the (symmetric) Fubini--Study metric. This conjecture is verified in all odd dimensions, for metrics on $d$-spheres when $d \neq 6$, for Riemannian manifolds satisfying the Raki\'c duality principle, and for K\"ahlerian manifolds.Comment: 33 page