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1. Conical Metrics on Riemann Surfaces, II: Spherical Metrics

   https://doi.org/10.1093/imrn/rnab011  

   Mazzeo, Rafe; Zhu, Xuwen (March 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We continue our study, initiated in [34], of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with some or all of the cone angles greater than $$2\pi $$. Deformations are obstructed precisely when the number $$2$$ lies in the spectrum of the Friedrichs extension of the Laplacian. Our main result is that, in this case, it is possible to find a smooth local moduli space of solutions by allowing the cone points to split. This analytic fact reflects geometric constructions in [37, 38]. 

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2. K3 metrics

   Kachru, Shamit; Tripathy, Arnav; Zimet, Max (June 2020, ArXivorg)

   null (Ed.)
3. Topological k-Metrics

   https://doi.org/10.4230/LIPIcs.SoCG.2024.13  

   Barkan, Willow; Bennett, Huck; Nayyeri, Amir (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   Metric spaces (X, d) are ubiquitous objects in mathematics and computer science that allow for capturing pairwise distance relationships d(x, y) between points x, y ∈ X. Because of this, it is natural to ask what useful generalizations there are of metric spaces for capturing "k-wise distance relationships" d(x_1, …, x_k) among points x_1, …, x_k ∈ X for k > 2. To that end, Gähler (Math. Nachr., 1963) (and perhaps others even earlier) defined k-metric spaces, which generalize metric spaces, and most notably generalize the triangle inequality d(x₁, x₂) ≤ d(x₁, y) + d(y, x₂) to the "simplex inequality" d(x_1, …, x_k) ≤ ∑_{i=1}^k d(x_1, …, x_{i-1}, y, x_{i+1}, …, x_k). (The definition holds for any fixed k ≥ 2, and a 2-metric space is just a (standard) metric space.) In this work, we introduce strong k-metric spaces, k-metric spaces that satisfy a topological condition stronger than the simplex inequality, which makes them "behave nicely." We also introduce coboundary k-metrics, which generalize 𝓁_p metrics (and in fact all finite metric spaces induced by norms) and minimum bounding chain k-metrics, which generalize shortest path metrics (and capture all strong k-metrics). Using these definitions, we prove analogs of a number of fundamental results about embedding finite metric spaces including Fréchet embedding (isometric embedding into 𝓁_∞) and isometric embedding of all tree metrics into 𝓁₁. We also study relationships between families of (strong) k-metrics, and show that natural quantities, like simplex volume, are strong k-metrics. 

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4. Snapshot Metrics Are Not Enough: Analyzing Software Repositories with Longitudinal Metrics

   https://doi.org/10.1145/3551349.3559517  

   Synovic, Nicholas M.; Hyatt, Matt; Sethi, Rohan; Thota, Sohini; Shilpika; Miller, Allan J.; Jiang, Wenxin; Amobi, Emmanuel S.; Pinderski, Austin; Läufer, Konstantin; et al (October 2022, Proceedings of the 37th IEEE/ACM International Conference on Automated Software Engineering)

   Software metrics capture information about software development processes and products. These metrics support decision-making, e.g., in team management or dependency selection. However, existing metrics tools measure only a snapshot of a software project. Little attention has been given to enabling engineers to reason about metric trends over time—longitudinal metrics that give insight about process, not just product. In thiswork,we present PRIME (PRocess MEtrics), a tool to compute and visualize process metrics. The currently-supported metrics include productivity, issue density, issue spoilage, and bus factor.We illustrate the value of longitudinal data and conclude with a research agenda. The tool’s demo video can be watched at https://bit.ly/ase2022-prime. Source code can be found at https://github.com/SoftwareSystemsLaboratory/prime. 

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5. Twisted Kähler–Einstein metrics

   https://doi.org/10.4310/PAMQ.2021.v17.n3.a8  

   Ross, Julius; Székelyhidi, Gábor (January 2021, Pure and Applied Mathematics Quarterly)

   null (Ed.)