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

A Combination of Teacher-Led Assessment and Self-Assessment Drives the Learning Process in Online Master Degree in Transplantation

Background: Good performance in a summative assessment does not always equate to educational gain following acourse. An educational programme may focus on improving student’s performance on a particular test instrument.For example, practicing multiple choice questions may lead to mastery of the instrument itself rather than testing theknowledge and its application. We designed an assessment strategy that consistently valid and reliable that would fitwith the students with a range of 27 nationalities with a different institutional, cultural and educational background inthis totally online masters programme in transplantation.Methods: Based on the published evidence, we analyzed 2 main assessment domains: (a) self-assessment and (b)peer-assessment. We compared them with traditional teacher-led assessment considering the diversity of students.Conclusion: We conclude that traditional teacher-led assessment supplemented by self-assessment is a strong drivein the learning process in this on-line course, whilst peer-assessment is challenging and associated with many flawsgiving the diversity of our students. Peer-assessment may be unreliable and not valid due to the difference in theinstitutional background and variation in experience between the students.</jats:p

Peripheral separability and cusps of arithmetic hyperbolic orbifolds

For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X-hyperbolic (n+1)-orbifold. A principal tool in the proof of this classification theorem is a subgroup separability result which may be of independent interest.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-32.abs.htm

Assessing Competence of NHS Consultants: Challenges and Possible Solutions

Even a mention of formal assessment of senior clinicians can be a contentious issue, to say least, when revalidation is said to be firmly in place in NHS-UK for almost half a decade. Since revalidation is accepted as a standard modality of assessment of performance, some colleagues in NHS wonder ‘stir up a hornets’ nest,’ when the authors allude to limitations of revalidation because poorly performing senior NHS clinicians may ‘slip through the net.’ NHS consultants have clinical as well as training roles. Fundamentally, this assessment (revalidation) is meant to ensure the safety of the public and mitigates the risk of disciplinary action by the GMC. Unfortunately, a disciplinary action is often the first sign of underperformance. In fact, the Bristol and Shipman inquiries have underscored the importance of the non-clinical and behavioural skills like communication, teamworking, personal organization and leadership are as important as clinical skills. Rather than considered an assessment tool, an annual appraisal is aimed to facilitate and improve the way NHS consultants work and provide services. The authors have to wait for five years, to assess the efficacy of the system that was introduced with much ‘fanfare’ since it was projected as a panacea for poor performance by ’bad doctors.’ The objectives of this article are to contextualize the issue of the underperformance among senior clinicians in the current NHS environment and to conceptualize the idea that their performance as trainers is directly related to their performance as clinicians. It is worth identifying the underlying factors of that are related to, or even better, can predict underperformance and will help evolve a strategy to help those consultants who are underperforming

Polynomial growth of volume of balls for zero-entropy geodesic systems

The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{pol}$} and the {\em weak polynomial entropy $h_{pol}^*$}. Both are infinite when the topological entropy is positive and they satisfy $h_{pol}^*\leq h_{pol}$. We first prove that the growth rate of the volume of balls is bounded above by means of the strong polynomial entropy and we show that for the flat torus this inequality becomes an equality. We then study the explicit example of the torus of revolution for which we can give an exact asymptotic equivalent of the growth rate of volume of balls, which we relate to the weak polynomial entropy.Comment: 22 page

Hyperbolic Geometry of Complex Networks

We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. We then establish a mapping between our geometric framework and statistical mechanics of complex networks. This mapping interprets edges in a network as non-interacting fermions whose energies are hyperbolic distances between nodes, while the auxiliary fields coupled to edges are linear functions of these energies or distances. The geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures. Finally, we show that targeted transport processes without global topology knowledge, made possible by our geometric framework, are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure

Fast approximation of centrality and distances in hyperbolic graphs

We show that the eccentricities (and thus the centrality indices) of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O(1)$ time an estimate $\hat{e}(v)$ of its eccentricity $ecc_G(v)$ such that $ecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ c\delta$ for a small constant $c$. We prove that every $\delta$-hyperbolic graph $G$ has a shortest path tree, constructible in linear time, such that for every vertex $v$ of $G$, $ecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ c\delta$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c'\delta$ can be computed in $O(|V|^2\log^2|V|)$ time, where $c'< c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author