Geometry of Hilbert schemes and the two numbers +1, -1

By work of Behrend, every complex scheme carries a canonical integer-valued constructible function, called the “Behrend function”.
In this talk, we will show that if a scheme has constant Behrend function, equal to ±1, then all of its components are generically reduced, and of the same dimension mod 2. This theoretical fact is tightly linked with: (1) Donaldson-Thomas theory, (2) expected pathologies on Quot schemes on 3-folds, (3) the parity conjecture for the Hilbert scheme of points on affine 3-space. This talk is about such links.