A bivariant Yoneda lemma (25 pages). This proves the universal property of the infinity-2-category of correspondences. (Some elements of a proof of this statement appear in part V of Gaitsgory and Rozenblyum’s book), but the methods there involve arcane manipulations of simplicial sets and are in any case incomplete.)

The universal property of derived geometry. Substantially revised version to appear by Christmas.

Here is a project proposal I wrote a while ago.

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.