  The main problem studied is resolution of singularities of the cotangent
sheaf of a complex- or real-analytic variety Y (or of an algebraic variety Y
over a field of characteristic zero). Given Y, we ask whether there is a global
resolution of singularities s: X -> Y such that the pulled-back cotangent sheaf
of Y is generated by differential monomials in suitable coordinates at every
point of X ("Hsiang-Pati coordinates''). Desingularization of the cotangent
sheaf is equivalent to monomialization of Fitting ideals generated by minors of
a given order of the logarithmic Jacobian matrix of s. We prove resolution of
singularities of the cotangent sheaf in dimension up to three. It was
previously known for surfaces with isolated singularities (Hsiang-Pati 1985,
Pardon-Stern 2001). Consequences include monomialization of the induced
Fubini-Study metric on the smooth part of a complex projective variety Y; there
have been important applications of the latter to L2-cohomology.
