Six operations in differential topology

The formalism of six operations, pioneered by Grothendieck and Verdier, serves as a unifying framework for studying cohomological phenomena. This language realizes Poincaré-type duality and transfer maps as certain adjunctions between stable $\infty$-categories of sheaves. In this talk, we highlight the theory of six operations in topology and apply it to provide an intrinsic version of the Pontryagin-Thom construction. We then discuss the intrinsic construction of invariants coming from Seiberg-Witten theory, which is based on the speaker's previous work.