In “Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow,” Velázquez constructed a countable collection of mean curvature flow solutions in ${\mathbb{R}}^N$ in every dimension $N\ge 8$. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. In contrast, we confirm here that, in every dimension $N\ge 8$, infinitely many of these solutions have uniformly bounded mean curvature.

Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix) due to the presence of “low-degree dependencies” such as isolated vertices and pairs of degree 1 vertices with the same neighborhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants $k\ge 3$ and $\mathit{\lambda }>0$, an Erdős–Rényi random graph $G\sim \mathbb{G}(n,\mathit{\lambda }∕n)$ with n vertices and edge probability $\mathit{\lambda }∕n$ typically has the property that its k-core (its largest subgraph with minimum degree at least k) is nonsingular. This resolves a conjecture of Vu from the 2014 International Congress of Mathematicians, and adds to a short list of known nonsingularity theorems for “extremely sparse” random matrices with density $O(1∕n)$. A key aspect of our proof is a technique to extract high-degree vertices and use them to “boost” the rank, starting from approximate rank bounds obtainable from (nonquantitative) spectral convergence machinery due to Bordenave, Lelarge, and Salez.

Borisov and Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi–Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localization of Edidin and Graham’s square root Euler class for $\mathrm{SO}(2n,\mathbb{C})$-bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localization formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a K-theoretic refinement by defining K-theoretic square root Euler classes and their localized versions. In a sequel, we prove that our invariants reproduce those of Borisov and Joyce.

We study the Du Bois complex ${\underset{\_}{\mathrm{\Omega }}}_Z^{\bullet }$ of a hypersurface Z in a smooth complex algebraic variety in terms of its minimal exponent $\tilde{\mathit{\alpha }}(Z)$. The latter is an invariant of singularities, defined as the negative of the greatest root of the reduced Bernstein–Sato polynomial of Z, and refining the log-canonical threshold. We show that if $\tilde{\mathit{\alpha }}(Z)\ge p+1$, then the canonical morphism ${\mathrm{\Omega }}_Z^p\to {\underset{\_}{\mathrm{\Omega }}}_Z^p$ is an isomorphism, where ${\underset{\_}{\mathrm{\Omega }}}_Z^p$ is the pth associated graded piece of the Du Bois complex with respect to the Hodge filtration. On the other hand, if Z is singular and $\tilde{\mathit{\alpha }}(Z)>p\ge 2$, we obtain non-vanishing results for some higher cohomologies of ${\underset{\_}{\mathrm{\Omega }}}_Z^{n-p}$.