We introduce persistence with an emphasis on its algebraic foundations, using the representation theory of posets. Linear representations of posets arise in several areas of mathematics, including the representation theory of quivers and finite dimensional algebras, Morse theory and other areas of geometry, as well as topological inference and topological data analysis -- often via persistent homology. In some of these contexts, the category of poset representations of interest admits a metric structure given by the so-called interleaving distance. Persistence studies the algebraic properties of these poset representations and their behavior under perturbations in the interleaving distance. We survey fundamental results in the area and applications to pure and applied mathematics, as well as theoretical challenges and open questions.

翻译：本文以偏序集的表示理论为核心，着重从代数基础角度介绍持久性理论。偏序集的线性表示出现在数学的若干领域，包括箭图与有限维代数的表示理论、莫尔斯理论及其他几何领域，以及拓扑推断与拓扑数据分析（通常通过持续同调实现）。在某些背景下，相关偏序集表示范畴可通过所谓的交错距离赋予度量结构。持久性理论研究这些偏序集表示的代数性质及其在交错距离扰动下的行为。我们综述该领域的基础性成果及其在纯数学与应用数学中的应用，并探讨理论挑战与开放性问题。