The Pangloss-Hirzebruch-Riemann-Roch theorem, II

The category of spectra indubitably is the best of possible worlds for cohomology theories. For instance in spectra one can start with a few basic morphisms, be confident that every natural diagram built from them will commute, and end up with a proof of the Hirzebruch-Riemann-Roch theorem. As in every good story we'll have a deus ex machina: Atiyah's identification of the Spanier-Whitehead dual of a manifold with the Thom spectrum of minus its tangent bundle. Based on joint work with Mattia Coloma and Eugenio Landi.