Classification of ancient noncollapsed flows in ${\mathbb{R}}^4$

In this paper, we classify all noncollapsed singularities of the mean curvature flow in ${\mathbb{R}}^4$. Specifically, we prove that any ancient noncollapsed solution either is one of the classical historical examples (namely ${\mathbb{R}}^j\times S^{3-j}$, $\mathbb{R}\times$2d-bowl, $\mathbb{R}\times$2d-oval, the rotationally symmetric 3d-bowl, or a cohomogeneity-one 3d-oval), or belongs to the 1-parameter family of ${\mathbb{Z}}_2\times {\mathrm{O}}_2$-symmetric 3d-translators constructed by Hoffman-Ilmanen-Martin-White, or belongs to the 1-parameter family of ${\mathbb{Z}}_2^2\times {\mathrm{O}}_2$-symmetric ancient 3d-ovals constructed by Du-Haslhofer. In light of the five prior papers on the classification program in ${\mathbb{R}}^4$ from our collaborations with Du, Hershkovits, and Choi-Daskalopoulos-Sesum, the major remaining challenge is the case of mixed behaviour, where the convergence to the round bubble-sheet is fast in $x_1$-direction, but logarithmically slow in $x_2$-direction. To address this, we prove a differential neck theorem, which allows us to capture the (dauntingly small) slope in $x_1$-direction. To establish the differential neck theorem, we introduce a slew of new ideas of independent interest, including switch and differential Merle-Zaag dynamics, anisotropic barriers, and propagation of smallness estimates. Applying our differential neck theorem, we show that every noncompact strictly convex solution is selfsimilarly translating, and also rule out exotic ovals.