With the dissemination of high-dimensional data, we wish to study its underlying geometry and answer long-debated questions in the fields that acquired the data. Although data can originate from various domains (e.g., images, surfaces), a reasonable assumption is that they lie on a manifold. At the heart of these challenges lies a fundamental question: how can we extract meaningful structure from thisdata? Real-world datasets are rarely clean - they are shaped by noise, outliers, and uneven sampling - and each new application introduces fresh scientific questions.

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