Is a donut the same as a coffee cup? Ask a topologist.
Topology is the study of those properties of objects that are not affected by continuous deformations. For example, properties such as stretching, bending and twisting, but not tearing. TDA is an emerging area in applied mathematics that combines elements of applied topology and computational geometry to analyze large data sets. It has been very successful in discovering structures and patterns in many large, complex and possibly noisy data sets by designing algorithms that exploit geometric and domain specific information about the “shape” of data.
One of the key messages in TDA is that data has shape and that shape matters.