Let $(M,g)$ be a complete Riemannian 3-manifold that is asymptotic to Schwarzschild with positive mass and whose scalar curvature vanishes. We unconditionally characterize the large, embedded stable constant mean curvature (CMC) spheres in $(M,g)$.

The uncovering of new structure on the Cuntz semigroup of a C*-algebra of stable rank one leads to several applications: we answer affirmatively, for the class of stable rank-one C*-algebras, a conjecture by Blackadar and Handelman on dimension functions, the global Glimm halving problem, and the problem of realizing functions on the cone of 2-quasitraces as ranks of Cuntz semigroup elements. We also gain new insights into the comparability properties of positive elements in C*-algebras of stable rank one.

A distinguishing feature of certain intractable problems in prime number theory is the sparsity of the underlying sequence. Motivated by the general problem of finding primes in sparse polynomial sequences, we give an estimate for the number of primes of the shape $x^3+2y^3$ where y is small.

We establish a compact analogue of the $P=W$ conjecture. For a projective irreducible holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total space. This builds a new connection between the topology of Lagrangian fibrations and the Hodge theory of hyper-Kähler manifolds. We present two applications of our result: one on the cohomology of the base and fibers of a Lagrangian fibration, and the other on the refined Gopakumar–Vafa invariants of a K3 surface. Furthermore, we show that the perverse filtration associated with a Lagrangian fibration is multiplicative under cup product.