The notion of covering type was recently introduced by Karoubi and Weibel to measure the complexity of a topological space by means of good coverings. When X has the homotopy type of a finite CW-complex, its covering type coincides with the minimum possible number of vertices of a simplicial complex homotopy equivalent to X. In this article we compute the covering type of all closed surfaces. Our results completely settle a problem posed by Karoubi and Weibel, and shed more light on the relationship between the topology of surfaces and the number of vertices of minimal triangulations from a homotopy point of view.2010 Mathematics Subject Classification. 57M20, 57Q15, 52B70, 57N16, 55M30, 55P15.

The growing adoption of decentralised renewable energy generation (such as solar photovoltaic panels and wind turbines) and low-carbon technologies will increase the strain experienced by the distribution networks in the near future. In such a scenario, energy storage is becoming a key alternative to traditional expensive reinforcements to network infrastructure, due to its flexibility, decreasing costs and fast deployment capabilities. In this work, an end-to-end data-driven solution to optimally design the control of a battery unit with the aim of reducing the peak electricity demand is presented. The proposed solution uses state-of-the-art machine learning methods for forecasting electricity demand and PV generation, combined with an optimisation strategy to maximise the use of photovoltaic energy to charge the energy storage unit. To this end, historical demand, weather, and solar energy generation data collected at the Stentaway Primary substation near Plymouth, UK, and at other six locations were employed.

We address the problem of estimating intrinsic distances in a manifold from a finite sample. We prove that the metric space defined by the sample endowed with a computable metric known as sample Fermat distance converges a.s. in the sense of Gromov-Hausdorff. The limiting object is the manifold itself endowed with the population Fermat distance, an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. This result is applied to obtain sample persistence diagrams that converge towards an intrinsic persistence diagram. We show that this method outperforms more standard approaches based on Euclidean norm with theoretical results and computational experiments.

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Electric load forecasting is becoming increasingly challenging due to the growing penetration of decentralised energy generation and power-electronics based loads such as heat pumps and electric vehicles, which adds to a transition to more variable work patterns (accentuated by the COVID-19 pandemic in 2020). In this paper, three different Machine Leaning models are analysed to predict the energy load one week ahead for a period of time including the COVID-19 pandemic. It is shown that, by using the recently proposed TabNet model architecture, it is possible to achieve an accuracy comparable to more traditional approaches based on gradient boosting and artificial neural networks without the need of performing complex feature engineering.