Topological Data Analysis (TDA) provides a powerful framework for extracting structure from complex data by studying its shape. This talk presents recent work on visualizing maps between high-dimensional spaces to detect correlations between datasets, alongside new adaptations of TDA to settings where representative sampling is impossible. This includes the integration of TDA with machine learning methodologies, particularly in contexts where traditional sampling is impractical, to analyze infinite datasets effectively.

A central theme is the application of these methods to knot theory, where the exponential growth in knot complexity places the space of knots and their invariants firmly in the realm of big data. Additional examples from cancer genomics and game theory highlight the broad applicability of these techniques across mathematics and the sciences.