Cohomology theories, ring stacks, and monoid stacks

• Deven Manam (University of Copenhagen)

Room: Amphithéâtre Yvonne Choquet-Bruhat Work of Simpson, Drinfeld, and Bhatt–Lurie gives a "stacky" approach to several known cohomology theories for schemes, providing a parsimonious way of encoding these theories and their coefficients in terms of quasicoherent sheaves on stacks; specifically, each theory is determined by a single ring stack, via the formalism of "transmutation". This perspective has been particularly illuminating in the prismatic/syntomic setting, where it exposes subtle arithmetic information in terms of geometry. I'll explain joint work with Dhilan Lahoti showing that the stacks arising here are in fact moduli spaces of ring stacks, and even moduli spaces of ring structures on certain monoid stacks. Viewed through the lens of transmutation, this result states that the theory of syntomification is "defined over F_1".

Can we count sheaves in positive characteristic ?

• Jiaqi FU (Université de Toulouse)

Room: Amphithéâtre Yvonne Choquet-Bruhat Donaldson--Thomas theory provides a virtual counting of coherent sheaves over complex Calabi–Yau 3-folds. From the deformation-theoretic perspective, DT invariants are shadow of derived critical locus structures. As a first step toward developing Donaldson–Thomas theory in positive characteristic, I study an enhanced notion of shifted symplectic structure using Antieau’s derived infinitesimal cohomology.