We give a cohomological classification of vector bundles of rank $2$ on a smooth affine threefold over an algebraically closed field having characteristic unequal to $2$. As a consequence we deduce that cancellation holds for rank $2$ vector bundles on such varieties. The proofs of these results involve three main ingredients. First, we give a description of the first nonstable ${\mathbb{A}}^1$-homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morel’s ${\mathbb{A}}^1$-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in ${\mathbb{A}}^1$-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.

We prove a black hole rigidity result for slowly rotating stationary solutions of the Einstein vacuum equations. More precisely, we prove that the domain of outer communications of a regular stationary vacuum is isometric to the domain of outer communications of a Kerr solution, provided that the stationary Killing vector-field $\mathbf{T}$ is small (depending only on suitable regularity properties of the black hole) on the bifurcation sphere. No other global restrictions are necessary.

The proof brings together ideas from our previous work with ideas from the classical work of Sudarsky and Wald on the staticity of stationary black hole solutions with zero angular momentum on the horizon. It is thus the first uniqueness result, in the framework of smooth, asymptotically flat, stationary solutions, which combines local considerations near the horizon, via Carleman estimates, with information obtained by global elliptic estimates.

We prove that equivariant multiplicities may be used to determine whether attractive fixed points on $T$-varieties are $p$-smooth, building on the Kumar–Arabia–Brion criterion for rational smoothness. In particular, this gives a combinatorial criterion for the determination of the $p$-smooth locus of Schubert varieties for all primes $p$.

A conjecture of Kadison and Kastler from 1972 asks whether sufficiently close operator algebras in a natural uniform sense must be small unitary perturbations of one another. For $n\ge 3$ and a free, ergodic, probability measure-preserving action of ${SL}_n\left(\mathbb{Z}\right)$ on a standard nonatomic probability space $(X,\mu )$, write $M=\left(L^{\infty }\right(X,\mu \left)⋊{SL}_n\right(\mathbb{Z}\left)\right)\overline{\otimes }R$, where $R$ is the hyperfinite II₁-factor. We show that whenever $M$ is represented as a von Neumann algebra on some Hilbert space $\mathcal{H}$ and $N\subseteq \mathcal{B}\left(\mathcal{H}\right)$ is sufficiently close to $M$, then there is a unitary $u$ on $\mathcal{H}$ close to the identity operator with $uM{u}^{\ast }=N$. This provides the first nonamenable class of von Neumann algebras satisfying Kadison and Kastler’s conjecture.

We also obtain stability results for crossed products $L^{\infty }(X,\mu )⋊\Gamma$ whenever the comparison map from the bounded to usual group cohomology vanishes in degree $2$ for the module $L^2(X,\mu )$. In this case, any von Neumann algebra sufficiently close to such a crossed product is necessarily isomorphic to it. In particular, this result applies when $\Gamma$ is a free group.

Let $A$ be a unital separable simple ${\mathrm{C}}^{\ast }$-algebra with a unique tracial state. We prove that if $A$ is nuclear and quasidiagonal, then $A$ tensored with the universal uniformly hyperfinite (UHF) algebra has decomposition rank at most one. We then prove that $A$ is nuclear, quasidiagonal, and has strict comparison if and only if $A$ has finite decomposition rank. For such $A$, we also give a direct proof that $A$ tensored with a UHF algebra has tracial rank zero. Using this result, we obtain a counterexample to the Powers–Sakai conjecture.

We conjecturally extract the triply graded Khovanov–Rozansky homology of the $(m,n)$ torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank $n-1$, and central character $m/n$. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov–Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to $q,t$-Catalan numbers and to previous conjectures of the last three authors relating knot homology to Hilbert schemes on singular curves.