PhD Thesis Defence

Speaker: Claudio Quadrelli (Western and Milan)

"The Structure of Absolute Galois Groups"

Time: 09:00

Room: MC 108

Analysis Seminar

Speaker: Edward Bierstone (University of Toronto)

" Hsiang-Pati coordinates"

Time: 14:30

Room: MC 107

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.)

Colloquium

Speaker: Rene Schoof (University of Rome, Tor Vergata)

"Elliptic curves and modular curves"

Time: 15:30

Room: MC 108

 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.