Schedule for 2018. 1. Modules on projective space. 2. Axioms for non-Archimedean geometry. 3. Geometric stacks.

Correspondences corepresent bivariant theories (10 pages). Probably already proved by Gaitsgory-Rozenblyum (part V of their book project), but in a way that uses rather more 4-fold simplicial sets than mine.

Here is a project proposal I wrote a while ago. At the moment, it appears that I will be ready to start work on it at the start of 2019.

Here is my theory of rigid analytic spaces based on monoids (125 pages). The point of it is to set up abstractly the well-known construction of Mumford-style toric degenerations from affine manifolds equipped with polyhedral decompositions. One thing that is missing from this (already rather long) paper is a discussion of moduli.

One section that might someday deserve its own paper defines *proper* morphisms in terms of extension properties. This has the advantage of not explicitly referring to closed sets, which are known to be a bit rubbish in monoid schemes. It is non-trivial (using Raynaud-Gruson flattening by blowups) to show that this definition is implied by Grothendieck’s definition.