Topological Data Analysis (TDA) has emerged as a powerful method to elucidate the “shape of data,” offering a promising toolset for handling highly dimensional and complex datasets. One of the key features of this methodology is to extract qualitative summaries based on connectivity information, generally independent of distance metric definitions or coordinate systems.

In this talk, I will explore applications of TDA on two very different examples: the distribution of galaxies across the universe and internal representations of large language models. Through these examples, I aim to demonstrate the versatility of TDA in addressing both the multiscale behavior inherent in static datasets, such as those in cosmology, and the temporal evolution characterizing dynamic systems like neural networks. This exploration highlights TDA’s potential to uncover intrinsic patterns and insights across diverse scientific domains.