Torus actions, Morse homology, and the Hilbert scheme of points on affine space

We formulate a conjecture on actions of the multiplicative group in motivic
homotopy theory. In short, if the multiplicative group G_m acts on a
quasi-projective scheme U such that U is attracted as t approaches 0 in G_m to
a closed subset Y in U, then the inclusion from Y to U should be an
A^1-homotopy equivalence.
We prove several partial results. In particular, over the complex numbers,
the inclusion is a homotopy equivalence on complex points. The proofs use an
analog of Morse theory for singular varieties. Application: the Hilbert scheme
of points on affine n-space is homotopy equivalent to the subspace consisting
of schemes supported at the origin.