Persistence originated from the observation that the critical values of a Morse function admit a canonical pairing inducing a direct sum decomposition of the sublevel set homology of the function into interval poset representations. The novel contribution of persistence is the stability result, which addresses questions such as: How can the critical values of a Morse function change when the function is perturbed? Initially motivated by geometric data science, persistence has since found applications in several areas of geometry and analysis. I will present an overview of the algebraic side of persistence, and describe recent work connecting two-dimensional Morse theory with the representation theory of the bigraded polynomial ring in two variables, shedding new light on these classical objects.