Joint IAS/PU Arithmetic Geometry

The Gysin map, or the wrong-way map, is a classical construction that has been available for various cohomology theories and in 𝐀¹-homotopy theory. In this talk, I will give a construction of it based on 𝐏¹-homotopy theory, so that it specializes to the Gysin map for all known or unknown cohomology theories with 𝐏¹-homotopy invariance, including étale, Hodge, crystalline, prismatic, etc. I will also survey applications of my Gysin map to more concrete subjects, including very recent works by Annala–Hoyois–Iwasa and Carmeli–Feng.