HS2026: 63133 Topological Data Analysis

Topology is the study of shape. Over the last two decades, a great deal of work has gone into applying topological ideas to problems in science and engineering, and above all to data analysis; this young field goes by several names, most often computational topology, applied topology, or topological data analysis (TDA). It sits at the intersection of topology, geometry, and algorithms, and its guiding question is how to make the shape of a data set precise, computable, and statistically meaningful. Geometric data is now everywhere, and much of it lives in high-dimensional spaces while being organized around lower-dimensional patterns and structures. TDA offers principled ways to detect, summarize, and compare such structure, and to feed it into pipelines for clustering, classification, and simplification.

This course surveys the central algorithms and techniques of TDA, covering both the theoretical foundations and the practical tools that are now in wide use across many domains. Along the way we will borrow from algebraic topology, geometry, linear and abstract algebra, algorithm design, statistics, and a little sheaf theory, building up to recent research results. We will study and use efficient software for the objects discussed in class, such as persistent homology and Reeb graphs, and we will look at applications in areas including computer graphics, image analysis, sensor networks, clustering, time series analysis, and genetics.

See https://elenawang93.github.io/TDA/

The prerequisites for this course are linear algebra and some programming experience

Englisch

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