Talk

This talk presents a purely topological model for wrapped Fukaya categories of surfaces, avoiding symplectic geometry and analytical techniques. The construction is based on combinatorial data of arcs and trajectories on surfaces, yielding a triangulated category that models the wrapped Fukaya category in a topological way. As an application, we obtain a homological mirror symmetry correspondence with gentle algebras arising from the same surface data.

This talk is based on joint work with Omar Gómez (Bielefeld) and Marius Nielsen (NTNU). Hopfological algebra is a variant of classical homological algebra introduced by Khovanov and Qi, motivated by potential applications in the categorification of quantum invariants of 3-manifolds. In this talk, I will explain an infinity-categorical approach to the theory that leads, in particular, to refined foundations as well as to "hopfological analogues" of classical invariants such as Hochschild (co)homology.

I will describe a stacky approach to syntomic cohomology of rigid spaces over $\mathbb{Q}_p$, which is an analytic analogue of the syntomification of a $p$-adic formal scheme of Drinfeld and Bhatt--Lurie. This yields a notion of syntomic cohomology of rigid spaces with coefficients which satisfies Poincaré duality, affords a theory of Chern classes and compares both to Hyodo--Kato and proétale cohomology.

    In the first part of the talk, I will explain what syntomic cohomology is, why one should care about a stacky approach and what the syntomification looks like geometrically. In the second part, I will show how one can use this geometric point of view to obtain cohomological comparison theorems.