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  "body": "\n## Summary\n\nTopological data analysis studies the shape of data with tools from algebraic topology.\nIt turns a finite set of points, usually in a [[metric space|0VKD1TP7F3]], into a family of [[simplicial complexes|V8GZ0W1ZZ5]] indexed by a scale.\nIt then records how topological features such as components and loops appear and disappear as the scale grows.\nThe main tool is [[persistent homology|S8SNBHVPSP]], whose output, a barcode or persistence diagram, is stable under small perturbations of the data.\nIt is used to find clusters, loops and voids in data from biology, materials science and image analysis.\n\n## Explanations\n\n### L1 human\n\nImagine points scattered around a ring.\nDraw a ball of radius $r$ around each point and watch what happens as $r$ grows.\nFor small $r$ the balls are separate dots; then they join into a ring with a hole; for large $r$ the hole fills in.\nThe hole lives for a long range of $r$, which is a strong sign that the data really lie on a circle.\nTopological data analysis measures such long-lived features and ignores the short-lived ones, which are mostly noise.\n",
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