ABSTRACT. We develop two new tools for use in Alexandrov geometry: a theory of ramified orientable double covers and a particularly useful version of the Slice Theorem for actions of compact Lie groups. These tools are applied to the classification of compact, positively curved Alexandrov spaces with maximal symmetry rank.

COVEN (Collaborative Virtual Environments) is a European project that seeks to develop a comprehensive approach to the issues in the development of collaborative virtual environment (CVE) technology. COVEN brings together twelve academic and industrial partners with a wide range of expertise in CSCW, networked VR, computer graphics, human factors, HCI, and telecommunications infrastructures. After two years of work, we are presenting the main features of our approach and results, our driving applications, the main components of our technical investigations, and our experimental activities. With different citizen and professional application scenarios as driving forces, COVEN is exploring the requirements and supporting techniques for collaborative interaction in scalable CVEs. Technical results are being integrated in an enriched networked VR platform based on the dVS and DIVE systems. Taking advantage of a dedicated Europe-wide ISDN and ATM network infrastructure, a large component of the project is a trial and experimentation activity that should allow a comprehensive understanding of the network requirements of these systems as well as their usability issues and human factors aspects.

Introduction: A discussion of 'waves' of the COVID-19 epidemic in different countries is a part of the national conversation for many, but there is no hard and fast means of delineating these waves in the available data and their connection to waves in the sense of mathematical epidemiology is only tenuous. Methods: We present an algorithm which processes a general time series to identify substantial, significant and sustained periods of increase in the value of the time series, which could reasonably be described as 'observed waves'. This provides an objective means of describing observed waves in time series. Results: The output of the algorithm as applied to epidemiological time series related to COVID-19 corresponds to visual intuition and expert opinion. Inspecting the results of individual countries shows how consecutive observed waves can differ greatly with respect to the case fatality ratio. Furthermore, in large countries, a more detailed analysis shows that consecutive observed waves have different geographical ranges. We also show how waves can be modulated by government interventions and find that early implementation of non-pharmaceutical interventions correlates with a reduced number of observed waves and reduced mortality burden in those waves. Conclusion: It is possible to identify observed waves of disease by algorithmic methods and the results can be fruitfully used to analyse the progression of the epidemic.

Oxford COVID-19 Database (OxCOVID19 Database) is a comprehensive source of information related to the COVID-19 pandemic. This relational database contains time-series data on epidemiology, government responses, mobility, weather and more across time and space for all countries at the national level, and for more than 50 countries at the regional level. It is curated from a variety of (wherever available) official sources. Its purpose is to facilitate the analysis of the spread of SARS-CoV-2 virus and to assess the effects of non-pharmaceutical interventions to reduce the impact of the pandemic. Our database is a freely available, daily updated tool that provides unified and granular information across geographical regions.

Let a compact Lie group act isometrically on a non-collapsing sequence of compact Alexandrov spaces with fixed dimension and uniform lower curvature and upper diameter bounds. If the sequence of actions is equicontinuous and converges in the equivariant Gromov-Hausdorff topology, then the limit space is equivariantly homeomorphic to spaces in the tail of the sequence. As a consequence, the class of Riemannian orbifolds of dimension n defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is finite up to orbifold homeomorphism.