Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau fourfolds

The Hilbert scheme of n points on a Calabi-Yau fourfold carries a virtual cycle of degree n constructed by Borisov-Joyce. I present a conjectural formula for the invariants obtained by capping the virtual cycle with the top Chern class of a tautological bundle. I discuss evidence for this conjecture by considering (1) small numbers of points and (2) toric fourfolds. For affine four-space, this implies a conjecture about solid partitions. The latter is a specialization of a recent conjecture on solid partitions derived in physics by N. Nekrasov. Joint work with Y. Cao.