Persistence Modules

This chapter introduces the basic properties of persistence modules. These can be defined over any partially ordered set; we are primarily interested in persistence modules over the real line or over a finite subset of the real line. In the best cases, a persistence module can be expressed as a direct sum of ‘interval modules’, the atomic building blocks of the theory. We introduce decorated real numbers to distinguish open and closed endpoints of real intervals. Not every persistence module is decomposable into interval modules, so we spend much of the monograph developing techniques that bypass this assumption. These techniques depend on a thorough understanding of certain finitely-indexed persistence modules known as A_n quiver representations. We develop the necessary algebraic tools, including a special ‘quiver calculus’ notation to facilitate the computation of numerical invariants of quiver representations.