  This is a review/announcement of results concerning the connection between
certain exactly solvable two-dimensional models of statistical mechanics,
namely loop models, and the equivariant $K$-theory of the cotangent bundle of
the Grassmannian. We interpret various concepts from integrable systems
($R$-matrix, partition function on a finite domain) in geometric terms. As a
byproduct, we provide explicit formulae for $K$-classes of various coherent
sheaves, including structure and (conjecturally) square roots of canonical
sheaves and canonical sheaves of conormal varieties of Schubert varieties.
