For any given finite subgroup $G\subset S{L}_3(\mathbb{C})$, we show that every projective crepant resolution X of the quotient variety ${\mathbb{C}}^3∕G$ is isomorphic to the moduli space of θ-stable G-constellations for a generic stability condition θ, as conjectured by Craw and Ishii. We also show that generators of the Cox ring of X can be obtained from semi-invariants for representations of the McKay quiver of G.

The Total Rank Conjecture is a coarser version of the Buchsbaum–Eisenbud–Horrocks conjecture which, loosely stated, predicts that modules with large annihilators must also have large syzygies. This conjecture was proved by the second author for rings of odd characteristic by taking advantage of the Adams operations on the category of perfect complexes with finite length homology. In this article, we prove a stronger form of the Total Rank Conjecture for rings of characteristic 2. This result may be seen as evidence that the Generalized Total Rank Conjecture (which is known to be false in odd characteristic) actually holds in characteristic 2. We also formulate a meaningful extension of the Total Rank Conjecture over non-Cohen–Macaulay rings and prove that this conjecture holds for any algebra over a field (independent of characteristic).

We study the Pauli operator in a 2-dimensional, connected domain with Neumann or Robin boundary condition. We prove a sharp lower bound on the number of negative eigenvalues reminiscent of the Aharonov–Casher formula. We apply this lower bound to obtain a new formula on the number of eigenvalues of the magnetic Neumann Laplacian in the semiclassical limit. Our approach relies on reduction to a boundary Dirac operator. We analyze this boundary operator in two different ways. The first approach uses Atiyah–Patodi–Singer (APS) index theory. The second approach relies on a conservation law for the Benjamin–Ono equation.

In this paper, we introduce the characteristic gluing problem for the Einstein vacuum equations. We present a codimension-10 gluing construction for characteristic initial data which are close to the Minkowski data and we show that the 10-dimensional obstruction space consists of gauge-invariant charges which are conserved by the linearized null constraint equations. By relating these 10 charges to the ADM energy, linear momentum, angular momentum, and the center of mass, we prove that asymptotically flat data can be characteristically glued (including the 10 charges) to the data of a suitably chosen Kerr spacetime, obtaining as a corollary an alternative proof of the Corvino–Schoen spacelike gluing construction. Moreover, we derive a localized version of our construction where the given data restricted on an angular sector is characteristically glued to the Minkowski data restricted on another angular sector. As a corollary, we obtain an alternative proof of the Carlotto–Schoen localized spacelike gluing construction. Our method yields no loss of decay in the transition region, resolving an open problem. We also discuss a number of other applications.