Atiyah duality and applications

In topology, Atiyah duality provides a geometric model for the dual of the suspension spectrum of a smooth manifold. In this talk, we export this into algebraic geometry by proving an analogous claim in the non-A^1-invariant stable motivic homotopy theory of Annala-Hoyois-Iwasa. Besides recovering many Poincaré duality type results, it has other surprising consequences to fundamental objects in algebraic geometry. I will explain how to use the so-called A^1-colocalization functor to prove the independence of logarithmic de Rham cohomology from the choice of good compactification in positive characteristic (without assuming resolution of singularities).