Recent developments in topological data analysis

Topological data analysis (TDA) employs various techniques from the mathematical field of topology to characterize the ‘’shape’’ of data. A key tool in TDA is persistent homology (PH), which, in its basic form, resembles hierarchical clustering in statistics while also capturing complex, higher-dimensional topological features like loops and voids. PH outputs a collection of intervals called the barcode, where each interval conveys the size and scale of a topological feature, making it valuable for analyzing data parameterized by a single parameter such as scale. Yet, many scenarios necessitate the simultaneous consideration of multiple parameters, such as when the data incorporates additional measurements or exhibits temporal variations. The transition from one to multiple variables presents formidable mathematical and algorithmic challenges, as such multiparameter persistent homology (MPH), is a highly active field of research, holding the promise to reshape TDA. In this talk, I will first give an introduction to TDA, and then discuss recent developments in MPH