Thoughts 2024-03-07:

Recently, I picked up some articles that introduced persistent homology to specific applicable fields across the discipline and read some of them. The basic idea of persistent homology is counting holes in a well-defined space. Such space, sometimes in a precise context, contained high-dimensional points which could refers to individuals with measured psychological index, countries with geometrical information, groups with highly complicated ideology. Those miscellaneous information emerged out of objects, they are dynamics, like liquid, and there are tensors between each. How to describe those relationships and how to classify the differences between spaces in this view is concerned. In other words, when you focus on topological information, you might lose a vision on local details.