AGNT Ical Atom

The Keller cylinder DG ring encodes homotopies between DG ring homomorphisms f_0, f_1 : A \to B.

Recently we discovered the higher cylinder DG rings Cyl_q(B), which assemble into the simplicial cylinder DG ring Cyl(B). For q=1 this recovers Keller’s original construction.

The sets SHom_q(A,B) of DG ring homomorphisms A \to Cyl_q(B) form the simplicial Hom set SHom(A,B). Our main result is that when A is a semi-free DG ring, the simplicial set SHom(A,B) is a Kan complex.

We prove several results about the fundamental groupoid SHom_{\leq 1}(A,B), including invariance under quasi-isomorphism B’ \to B, and that the automorphism groups are abelian. We also indicate some applications of this work.

Typed notes: https://drive.google.com/file/d/1sMzwoC_DGCotOfak8o8wYpmttgZELf6l/view

arXiv eprint: https://arxiv.org/abs/2602.11943

In this talk I will present some enhancements and generalizations of a criterion for six-functor formalisms first sketched by Voevodsky in 2001. This principle was then implemented by Ayoub in order to show that the stable motivic homotopy theory of quasi-projective schemes has the structure of a six-functor formalism, although it has later been generalized by works of Cisinski, Déglise, Hoyois, Khan, and Ravi, leading to a six-functor formalism of genuine stable motivic homotopy theory on qcqs derived algebraic stacks with separated diagonals and nice stabilizers.

In our framework, we produce six-functor formalism using the cohomological behaviour of smooth maps, closed immersions, and smooth proper maps (where the relevant cohomological property is expressed by a version of Atiyah duality). This is related to recent results of Dauser-Kuijper and Cnossen-Lenz-Linskens, which enhances work of Mann following Liu-Zheng on the construction of six-functor formalisms using the cohomological behaviour of étale maps and proper maps. Our general results are then used to produce a six-functor formalism of complex analytic stable motivic homotopy theory, as well as equivariant analytification functors that are compatible with the six operations.

The discovery of the Fargues–Fontaine curve has led to major advances in the geometrization of $p$-adic Hodge theory. In this talk, we explain how several $p$-adic cohomology theories can be realized as vector bundles on the Fargues–Fontaine curve. We then present a motivic approach to show the uniqueness of such vector bundles, which, in particular, yields comparison theorems for them. Moreover, we show that one can choose a canonical comparison isomorphism between these vector bundles.