A motivation is to avoid black box thinking when it comes to making conclusions from data. We should not be satisfied with the “what” but also ask the “how” and “why”. There are also a myriad of ethical issues that can occur when we use biased data and then perpetuate problems because of data analysis techniques where we cannot interpret the results. To properly analyze large and complicated data we should incorporate the tools from mathematics. We can construct, theoretically study, efficiently implement, and apply data analysis methods which incorporate ideas from pure mathematics. 

Research in the field of Topological Data Analysis uses tools from geometry and topology to understand the shape the shape of data. We can process raw data into topological measurements through which we can detect and quantify structures within data. The analysis of these topological measurements can be mathematically sophisticated and statistically rigorous, but also have the potential for inference and interpretation that is lacking in many current machine learning and AI methods.