Which "topological" objects (broadly construed) are defined by polynomial equations? Attempts to answer this question have motivated much of algebraic and arithmetic geometry over the last hundred years. For example, the well-known Hodge and Tate conjectures give proposals for how to do so, in terms of the complex geometry and arithmetic of a variety, respectively. I'll discuss non-abelian analogues of these conjectures, due to Simpson and Fontaine-Mazur, and some results towards them on generic curves, obtained jointly with Aaron Landesman. These results, which resolve conjectures of Esnault-Kerz and Budur-Wang and answer questions of Kisin and Whang, ultimately boil down to very concrete group-theoretic statements, whose proofs perhaps surprisingly rely on non-abelian Hodge theory, low-dimensional topology, and the Langlands program.