I'll start with some background about K-cohomology and K-homology of algebraic varieties, e.g. Poincar\'e duality, cup, and cap products. Unlike ordinary cohomology, it generalizes to the equivariant case without passing through infinite dimensions.
I'll talk about two particularly nice ways to specify some elements of K-homology: subschemes and branchvarieties. In either case the K-analogue of homotopy is provided by "flat families". I'll explain why flat families of subschemes/branchvarieties of projective space have constant Hilbert polynomial, which one can identify with the element of K(projective space).
I'll recall the standard fact that any 1-parameter flat family of projective varieties defined for all nonzero t can be uniquely extended across t=0 to a flat family of subschemes. Unfortunately this can produce "nonreduced" subschemes, so I'll state a theorem from last year, that such families can also be extended (differently) to a flat family of branchvarieties.
Then I'll get to the only new theorem in this talk: a criterion for the limit subscheme to equal the limit branchvariety. Time permitting, I'll use it to compute the equivariant K-classes of Schubert varieties in flag manifolds.