In this thesis, we apply cochain complexes as an algebraic model of space in a diverse range of mathematical and scientific settings. We begin with an algebraic-discrete Morse theory model of auto-encoding cochain data, connecting the homotopy theory of deformation retracts with the loss function. We then make use of the multiplicative structure of cochain complexes and PL differential forms in generalizing rational homotopy theory to the persistent setting, translating the well-known structure theorems from Topological Data Analysis (TDA) into cell decompositions and Postnikov towers in the respective categories of persistent CDGAs and copersistent spaces. The integration theory of differential forms is then used to define a robust representation learning framework for simplicial complexes embedded in Euclidean space. Finally, the Hodge theory of differential forms and their associated simplicial cochains are used to model closed biological processes in the RNA transcriptome and to estimate the corresponding cascade of driver genes.