Absolute profinite rigidity and hyperbolic geometry

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[\omega])$ with $\omega^2+\omega+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).Comment: v2: 35 pages. Final version. To appear in the Annals of Mathematics, Vol. 192, no. 3, November 202

Probabilistic computer model of optimal runway turnoffs

Landing delays are currently a problem at major air carrier airports and many forecasters agree that airport congestion will get worse by the end of the century. It is anticipated that some types of delays can be reduced by an efficient optimal runway exist system allowing increased approach volumes necessary at congested airports. A computerized Probabilistic Runway Turnoff Model which locates exits and defines path geometry for a selected maximum occupancy time appropriate for each TERPS aircraft category is defined. The model includes an algorithm for lateral ride comfort limits

Stable de Sitter vacua in N=2, D=5 supergravity

We find 5D gauged supergravity theories exhibiting stable de Sitter vacua. These are the first examples of stable de Sitter vacua in higher-dimensional (D>4) supergravity. Non-compact gaugings with tensor multiplets and R-symmetry gauging seem to be the essential ingredients in these models. They are however not sufficient to guarantee stable de Sitter vacua, as we show by investigating several other models. The qualitative behaviour of the potential also seems to depend crucially on the geometry of the scalar manifold.Comment: 26 pages, v2:typos corrected, published versio

Counting and effective rigidity in algebra and geometry

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.Comment: v.2, 39 pages. To appear in Invent. Mat

Conformational Reorganization of the SARS Coronavirus Spike Following Receptor Binding: Implications for Membrane Fusion

The SARS coronavirus (SARS-CoV) spike is the largest known viral spike molecule, and shares a similar function with all class 1 viral fusion proteins. Previous structural studies of membrane fusion proteins have largely used crystallography of static molecular fragments, in isolation of their transmembrane domains. In this study we have produced purified, irradiated SARS-CoV virions that retain their morphology, and are fusogenic in cell culture. We used cryo-electron microscopy and image processing to investigate conformational changes that occur in the entire spike of intact virions when they bind to the viral receptor, angiotensin-converting enzyme 2 (ACE2). We have shown that ACE2 binding results in structural changes that appear to be the initial step in viral membrane fusion, and precisely localized the receptor-binding and fusion core domains within the entire spike. Furthermore, our results show that receptor binding and subsequent membrane fusion are distinct steps, and that each spike can bind up to three ACE2 molecules. The SARS-CoV spike provides an ideal model system to study receptor binding and membrane fusion in the native state, employing cryo-electron microscopy and single-particle image analysis