Topological invariants arising from deformations of spaces into matrices and operator algebras

Starting with elementary examples, we shall discuss how deformations into matrix algebras of the commutative algebra of continuous functions on a space X relate to the K-homology of X. Along the way we shall illustrate by examples the contrast between homotopy and asymptotic homotopy. If time allows, we plan to further elaborate these ideas in the context of asymptotic morphisms of Connes and Higson and continuous fields of operator algebras.