Unlike the classical tools and methods that data scientists have historically employed to analyze data, topological data analysis (TDA) provides a tool for analyzing the shape of data. TDA was first inspired by the work of Marston Morse, and the foundations of the field were laid out by Edelsbrunner, Letscher, and Zomorodian in 2002 and Carlsson in 2005. One of the central tasks in TDA is finding the proper representation of a given data cloud. Such representation contains algebraic and geometric features of the data cloud and one can understand the shape of the data using those features. Moreover, one can evaluate the difference or similarity of two data clouds by measuring the similarity of their barcodes.

The theory of 1-dimensional TDA was developed in the two decades following the beginning of the field. Although the theory of 1-dimensional TDA is a powerful tool helping people understand data, the data cloud itself may naturally contain more than one parameter. Analyzing one parameter at a time is not only time-consuming but also information-losing: because taking one parameter at a time only gives a slice of the feature by fixing the other parameters. Multi-parameter topological data analysis has promising application potential in dealing with higher-dimensional data clouds, however, the theory of multi-parameter TDA is currently under development. It is known that, when there are multiple parameters, it is impossible to obtain a classification theorem, and such a theorem is crucial in order to generalize the notion of barcodes to the higher dimensions.

My current project is to refine a collection of objects in the category of multipersistence modules so that the obtained subcategory is as large as possible while the analogs of the classification theorem of one parameter hold on this subcategory.