*Abstract:* In this age of high-dimensional data, many challenging questions take the following shape: can you check whether the data has a certain desired property by checking that property for many, but low-dimensional data fragments? In recent years, such questions have inspired new, exciting research in commutative algebra, especially relevant when the property is highly symmetric and expressible through systems of polynomial equations.I will discuss three concrete questions of this kind that we have settled in the affirmative: Gaussian factor analysis from an algebraic perspective, high-dimensional tensors of bounded rank, and higher secant varieties of Grassmannians. The theory developed for these examples deals with group actions on infinite-dimensional algebraic varieties, and applies in a much wider context. In particular, I will sketch its relation to the fantastic Matroid Minor Theorem due to Geelen, Gerards, and Whittle.