Given a mod $p$ Galois representation, one often wonders whether it arises by reducing a $p$-adic one, and whether these lifts are suitably 'well-behaved'. In this talk, we discuss how ideas from homotopy theory aid the study of Galois deformations, reviewing work of Galatius-Venkatesh.

The arithmetic regularity lemma says that any dense set A in F_p^n can be cut along cosets of some small codimension subspace H <= F_p^n such that on almost all cosets of H, A is either random or structured (in a precise quantitative manner). A standard example shows that one cannot hope to improve "almost all" to "all", nor to have a good quantitative dependency between the constants involved. Adding a further combinatorial assumption on A to the arithmetic regularity lemma makes its conclusion so strong that one can essentially classify such sets A. In this talk, I will use use the analogous problem with F_p^n replaced with R^n as a way the motivate the funny title.