ABSTRACT: This lecture will examine the t^r conditions which were
introduced by Thom and Trotman, which concern families of C^r
sections of a singular set. For real and complex analytic sets, we
show that the t^r conditions have algebraic formulations in terms
of integral closure of modules. Our formulation gives a new simple
proof, for analytic sets, of the change in the conditions under
Grassmann modification proved by Kuo and Trotman for subanalytic sets;
this is used in conjunction with the principle of specialization of
integral dependence to give numerical criteria for familes of plane
sections of complex complete intersections to be Whitney equisingular.