We define analytic torsion for the twisted de Rham complex, consisting of differential forms on a compact Riemannian manifold X with coefficients in a flat vector bundle E, with a differential given by a flat connection on E plus a closed odd degree differential form on X. The definition in our case is more complicated than in the case discussed by Ray-Singer, as it uses pseudodifferential operators. We show that this analytic torsion is independent of the choice of metrics on X and E, establish some basic functorial properties, and compute it in many examples. We also establish the relationship of an invariant version of analytic torsion for T-dual circle bundles with closed 3-form flux. This is joint work with Siye Wu.

I shall mix classical analyis and modern computation to study integrals of powers of sinc functions and related objects. Along the way I shall demonstrate some remarkable false identities and some almost-as-remarkable true ones. This talk is based on joint research with various researchers aged eighteen (Bernard Mares) to eighty-five (David Borwein).