In algebraic topology, pushforwards in generalized cohomology theories arise from a single geometric construction: the Thom collapse map. This talk explains an algebro-geometric analogue of that construction. Although special cases have long been available, a general Thom collapse map in algebraic geometry has only recently been developed by Longke Tang in the setting of non-A^1-invariant motivic homotopy theory introduced by Annala, Hoyois, and Iwasa. Tang’s construction gives a universal source of pushforwards for essentially all cohomology theories occurring in algebraic geometry.
I will explain Tang's construction, the subtleties that are present in the non-A^1-invariant setting, and how this perspective gives a conceptual explanation for many standard properties of cohomology theories. If time permits, I will discuss an application to logarithmic geometry: the cohomology of nice enough log schemes can be reformulated in terms of cohomology of ordinary schemes.