  We describe here some recent progress pertaining to the Serre Intersection
Multiplicity Conjecture. In particular, we show that if A is an unramified
regular local ring, then just as in the equicharacteristic case, the
intersection multiplicity of two complimentary-dimensional modules is bounded
below by the product of their Hilbert-Samuel multiplicities. We also explain,
in terms of the blowup of Spec(A) at the closed point, the geometric
significance of achieving this lower bound.
