We present a complete generalization of Kirwan’s partial desingularization theorem on quotients of smooth varieties. Precisely, we prove that if $\mathcal{X}$ is an irreducible Artin stack with stable good moduli space $\mathcal{X}\stackrel{\mathit{\pi }}{\to }\mathbf{X}$, then there is a canonical sequence of birational morphisms of Artin stacks ${\mathcal{X}}_n\to {\mathcal{X}}_{n-1}\to \cdots \to {\mathcal{X}}_0=\mathcal{X}$ with the following properties: (1) the maximum dimension of a stabilizer of a point of ${\mathcal{X}}_{k+1}$ is strictly smaller than the maximum dimension of a stabilizer of ${\mathcal{X}}_k$ and the final stack ${\mathcal{X}}_n$ has constant stabilizer dimension; (2) the morphisms ${\mathcal{X}}_{k+1}\to {\mathcal{X}}_k$ induce projective and birational morphisms of good moduli spaces ${\mathbf{X}}_{k+1}\to {\mathbf{X}}_k$. If in addition the stack $\mathcal{X}$ is smooth, then each of the intermediate stacks ${\mathcal{X}}_k$ is smooth and the final stack ${\mathcal{X}}_n$ is a gerbe over a tame stack. In this case the algebraic space ${\mathbf{X}}_n$ has tame quotient singularities and is a partial desingularization of the good moduli space X. When $\mathcal{X}$ is smooth our result can be combined with D. Bergh’s recent destackification theorem for tame stacks to obtain a full desingularization of the algebraic space X.

A well-known conjecture states that a random symmetric $n\times n$ matrix with entries in $\{-1,1\}$ is singular with probability $\mathrm{\Theta }(n^2{2}^{-n})$. We prove that the probability of this event is at most $exp(-\mathrm{\Omega }(\sqrt{n}))$, improving the best-known bound of $exp(-\mathrm{\Omega }(n^{1∕4}\sqrt{logn}))$, which was obtained recently by Ferber and Jain. The main new ingredient is an inverse Littlewood–Offord theorem in ${\mathbb{Z}}_p^n$ that applies under very mild conditions, whose statement is inspired by the method of hypergraph containers.

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Several noteworthy examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twist flats for mapping class groups; the maximal rank of a free abelian subgroup for right-angled Coxeter groups and right-angled Artin groups (in the latter this can also be observed as the clique number of the defining graph); and, for the Weil–Petersson metric, the rank is the integer part of half the complex dimension of Teichmüller space.

We prove that, in a hierarchically hyperbolic space (HHS), any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a very mild condition on the HHS satisfied by all natural examples). This resolves outstanding conjectures when applied to a number of different groups and spaces.

In the case of the mapping class group, we verify a conjecture of Farb. For Teichmüller space we answer a question of Brock. In the context of certain $\mathrm{CAT}(0)$ cubical groups, our result handles novel special cases, including right-angled Coxeter groups.

An important ingredient in the proof, which we expect will have other applications, is that the hull of any finite set in an HHS is quasi-isometric to a $\mathrm{CAT}(0)$ cube complex of dimension bounded by the rank. (If the HHS is a $\mathrm{CAT}(0)$ cube complex, then the rank can be lower than the dimension of the space.)

We deduce a number of applications of these results. For instance, we show that any quasi-isometry between HHSs induces a quasi-isometry between certain factored spaces, which are simpler HHSs. This allows one, for example, to distinguish quasi-isometry classes of right-angled Artin/Coxeter groups.

Another application of our results is to quasi-isometric rigidity. Our tools in many cases allow one to reduce the problem of quasi-isometric rigidity for a given hierarchically hyperbolic group to a combinatorial problem. As a template, we give a new proof of quasi-isometric rigidity of mapping class groups, which, once we have established our general quasiflats theorem, uses simpler combinatorial arguments than in previous proofs.

We present new estimates for sums of the divisor function and other similar arithmetic functions in short intervals over function fields. (When the intervals are long, one obtains a good estimate from the Riemann hypothesis.) We obtain an estimate that approaches square-root cancellation as long as the characteristic of the finite field is relatively large. This is done by a geometric method, inspired by work of Hast and Matei, where we calculate the singular locus of a variety whose ${\mathbb{F}}_q$-points control this sum. This has applications to highly unbalanced moments of L-functions.

We construct examples of solutions to the conservative surface quasigeostrophic (SQG) equation that must either exhibit infinite-in-time growth of derivatives or blow up in finite time.