Let X be a higher-rank symmetric space or a Bruhat–Tits building of dimension at least 2 such that the isometry group of X has property $(T)$. We prove that for every torsion-free lattice $\mathrm{\Gamma }\subset \mathrm{Isom}X$, any homology class in $H_1(\mathrm{\Gamma }\setminus X,{\mathbb{F}}_2)$ admits a representative cycle of total length $o_X(\mathrm{Vol}(\mathrm{\Gamma }\setminus X))$. As an application, we show that ${dim}_{{\mathbb{F}}_2}{H}_1(\mathrm{\Gamma }\setminus X,{\mathbb{F}}_2)=o_X(\mathrm{Vol}(\mathrm{\Gamma }\setminus X))$.

Dave Gabai recently proved a smooth 4-dimensional “light bulb theorem” in the absence of 2-torsion in the fundamental group. We extend his result to 4-manifolds with arbitrary fundamental group by showing that an invariant of Mike Freedman and Frank Quinn gives the complete obstruction to “homotopy implies isotopy” for embedded 2-spheres which have a common geometric dual. The invariant takes values in an ${\mathbb{F}}_2$-vector space generated by elements of order 2 in the fundamental group and has applications to unknotting numbers and pseudoisotopy classes of self-diffeomorphisms. Our methods also give an alternative approach to Gabai’s theorem using various maneuvers with Whitney disks and a fundamental isotopy between surgeries along dual circles in an orientable surface.

We consider the Anderson model with Bernoulli potential on the three-dimensional (3D) lattice ${\mathbb{Z}}^3$, and prove localization of eigenfunctions corresponding to eigenvalues near zero, the lower boundary of the spectrum. We follow the framework of Bourgain–Kenig and Ding–Smart, and our main contribution is a 3D discrete unique continuation, which says that any eigenfunction of the harmonic operator with bounded potential cannot be too small on a significant fractional portion of all the points. Its proof relies on geometric arguments about the 3D lattice.

Let $(M,\mathit{\omega })$ be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu–Yau and Tosatti–Yang that M is necessarily projective and has ample canonical bundle. In this paper, we show that any irreducible subvariety of M is of general type, thus confirming in this particular case a celebrated conjecture of Lang. Moreover, we can extend the theorem to the quasinegative curvature case building on earlier results of Diverio–Trapani. Finally, we investigate the more general setting of a quasiprojective manifold $X^{\circ }$ endowed with a Kähler metric with negative holomorphic sectional curvature, and we prove that such a manifold $X^{\circ }$ is necessarily of log-general type.

We show that the blowups of compact solutions to the mean curvature flow in ${\mathbb{R}}^N$ initially satisfying the pinching condition $|A{|}^2<c|H{|}^2$ for a suitable constant $c=c(n)$ must be codimension one.