13.11

Wednesday, 13 November, 16-18 MEB331

Hilb T*C and Higgs bundles, by Danielle Boccalini

This talk is a trial for a talk that I will give in a graduate workshop on Hilbert Schemes. Every comment is very welcome.

We will discuss analogies between the moduli space M of Higgs bundles over a curve C and the Hilbert scheme Hilb T*C of points on the cotangent bundle of the curve C. An Higgs field over a curve C is the data of a vector bundle and a twisted endomorphism of it. Taking the characteristic polynomial of the endomorphism gives a proper Lagrangian fibration of M, whose generic fiber is a Jacobian variety. This leads to a birational map from M to Hilb T*C. We will concentrate on the case where C is an elliptic curve. Here things are better behaved, and, taking into account a parabolic structure on the Higgs bundles, it is possible to extend the birational map to an actual isomorphism. This is achieved thanks to a Fourier-Mukai transform analysis. The talk is based on Nakajima's book and on an article of M. Groechenig http://arxiv.org/pdf/1206.5516.pdf

Parallel transport for Fuchsian connections, by Szilard Szabo.

We will discuss the iterated integral expansion of parallel transport and regularised parallel transport maps for Fuchsian connections, and study the change of these maps under suitable modifications of the path. In particular, we deduce a formula relating polylogarithms corresponding to various paths.