Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .

We prove versions of various classical results on specialisation of fundamental groups in the context of log schemes in the sense of Fontaine and Illusie, generalising earlier results of Hoshi, Lepage and Orgogozo. The key technical result relates the category of finite Kummer étale covers of an fs log scheme over a complete Noetherian local ring to the Kummer étale coverings of its reduction.

We give a construction of the moduli space of stable maps to the classifying stack Bμr of a cyclic group by a sequence of rth root constructions on . We prove a closed formula for the total Chern class of μr-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus-zero Gromov–Witten theory of stacks of the form [ℂN/μr]. We deduce linear recursions for genus-zero Gromov–Witten invariants.

For a Whitney stratification S of a space X (or, more generally, a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an S-constructible stack of categories on X. The motivating example is the stack of S-constructible perverse sheaves. We introduce a 2-category EP≤2(X,S), called the exit-path 2-category, which is a natural stratified version of the fundamental 2-groupoid. Our main result is that the 2-category of S-constructible stacks on X is equivalent to the 2-category of 2-functors 2Funct(EP≤2(X,S),Cat) from the exit-path 2-category to the 2-category of small categories.

Lie algebroids cannot always be integrated into Lie groupoids. We introduce a new structure, ‘Weinstein groupoid’, which may be viewed as stacky groupoids. We use this structure to present a solution to the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie algebroid can be integrated into such a groupoid.