Abstract: Given a complex analytic (or algebraic) variety X, can we find a resolution of singularities p: Y -> X such that the pulled-back cotangent sheaf is generated by differential monomials in suitable coordinates at every point of Y ("Hsiang-Pati coordinates")? The answer is "yes" in dimension up to 3. It was previously known for surfaces X 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 X. In the case of a surface X, Hsiang and Pati used this to prove that the intersection cohomology of X (with the middle perversity) equals the $L^2$ cohomology of the smooth part of X (Cheeger-Goresky- Macpherson conjecture). Existence of Hsiang-Pati coordinates is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of p. (Joint work with Andre Belotto, Vincent Grandjean and Pierre Milman.)

Abstract:  In 1972 J.-P. Serre proved an important theorem concerning the Galois action on the torsion points of elliptic curves over number fields. We describe a conjectural uniform version of this theorem and its relation to rational points on modular curves.