ABSTRACT: Commutative rings often appear as rings of functions on geometricallydefined objects, such as smooth varieties or algebraic varieties. In this context,commutative algebras model morphisms of the underlying varieties, and homology theories of commutative algebras measure the singularities of the fibers of such morphisms.
At the beginning of the talk it will be explained how classical results linkimportant properties of commutative algebras, such as smoothness, to the vanishing of homology groups in the theories constructed by Gerhard Hochschild in the mid-1940's,and by Michel Andr\'e and Daniel Quillen in the late 1960's. This will be followedby a discussion of new homological characterizations of several classes of algebras in terms of vanishing of homology groups or finiteness of homology algebras. Theseresults come partly from joint work with Srikanth Iyengar. Their proofs use non-trivial connections with various topics in algebraic topology, such as rational homotopy theory or thehomology of Eilenberg-MacLane spaces.