Mathematics and worked lessons

A persistence module keeps both its spaces and the maps relating them. A finite encoding makes that description computable without discarding the information carried by those maps. These lessons develop the mathematics through definitions, examples and interpreted computations.

The reading map offers example-first and definitions-first entrances. It shows useful continuations, while this collection lets you open a treatment directly. Install TamerOp when you want to run a lesson; the mathematical explanations and saved figures can be read without a Julia session.

• A hole appears and disappears — Mathematical lesson. When does a hole appear and disappear, and how can I read its interval?
• Why two parameters change the problem — Mathematical lesson. Why do two parameters require spaces and maps beyond a barcode?
• Persistence modules over posets — Mathematical lesson. How do vector spaces and compatible maps describe persistence?
• Finite encodings: recovering a module from finite data — Mathematical lesson. How can finite data recover a module over an infinite parameter domain?
• Inspect spaces and maps — Mathematical lesson. How does the computed encoding recover the square and its spaces and maps?
• Indicator presentations: building a module from regions and a matrix — Mathematical lesson. How can regions and a matrix define spaces and their maps?
• Tameness and scope: when finite descriptions exist — Mathematical lesson. When do constant subdivisions, finite encodings and finite fringe presentations exist?
• Why finite computations stay tame — Mathematical lesson. Why do finite constructions and compatible maps preserve tame descriptions?
• Categories, encodings, and derived computations — Mathematical lesson. In which category does a computation take place, and when can its result be transported?

For a workflow using an object you already understand, consult Using TamerOp. The topic map brings together lessons, usage, precise contracts, implementation accounts and measured comparisons around shared subjects.