I will give an update on a proposed model theory for directed limits and colimits of first-order structures, originally motivated by applications to commutative algebra and the model theory of valued fields. To illustrate the usefulness of the formalism, I will prove a new general AKE theorem in mixed characteristic in a language with a cross-section of the value group and a lift of the residue field.

I will also discuss connections with other approaches to this topic, including pro- and ind-definable sets, infinitary logic, Feferman's local functors, accessible functors, and ultraproducts, some of which I have not discussed previously.

There has been substantial progress in the construction of eigenvarieties and $p$-adic families of automorphic forms, and their relationship with Selmer groups and ($p$-adic) $L$-functions. In this talk I will introduce some of these constructions, starting with modular forms, and the concept of complete $p$-adic rigidity: the non-existence of nontrivial $p$-adic deformations. I will explain some of the techniques used to study the geometry of eigenvarieties, and how these specialise to show that certain noncuspidal 'Saito—Kurokawa' points are completely $p$-adically rigid. If time permits, I will also briefly outline how similar strategies may be used to construct $p$-adic families through cuspidal, nonholomorphic Saito—Kurokawa points and to produce nontrivial Selmer classes predicted by the Bloch—Kato conjecture.