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# Motivic homotopy theory beyond A^1-invariance

## June 1, 2023 at 14:15 – 15:15 CEST

Ryomei Iwasa (Paris)

Abstract: The basic question I’d like to address in this talk is the following: How to do homotopy theory in algebraic geometry while keeping the affine line A^1 non-contractible? I will explain that tensor invertibility of the pointed projective line P^1 supplies homotopies between projective bundle sections in a non-trivial but canonical way. This dramatically expands the scope of motivic homotopy theory, and non-A^1-invariant theories such as syntomic cohomology, prismatic cohomology, algebraic K-theory, and topological cyclic homology can be studied from this perspective. In particular, I’ll explain that algebraic and Selmer K-theory are described by Snaith-type formulas. Based on joint work with Toni Annala and Marc Hoyois.