I think about:

Non-archimedean periods for log Calabi-Yau surfaces

Abstract: Period integrals are a fundamental concept in algebraic geometry and number theory. Using the non-archimedean SYZ fibration, we study the notion of non-archimedean periods as introduced by Kontsevich and Soibelman. We prove that the non-archimedean period map recovers the analytic period for log Calabi-Yau surfaces, verifying a conjecture of Kontsevich-Soibelman. This is joint work with Jonathan Lai.

Analytification of logarithmic schemes

Abstract: We construct an analytification functor which, to a log scheme (X,MX) over a valued field, associates a locally ringed space (X,MX)an. This formalises the notion of that the Berkovich analytification of X is equal to the tropicalisation of (X,OX). The functor passes through the category of valuative log schemes, and therefore we obtain an explicit description of the valuativisation of a log scheme in the trivially valued case. As a result, any functor which is invariant under log modifications induces an associated functor on the corresponding analytic spaces. We offer a detailed description in the case of LogChow.

Furthermore, to each log scheme (X,MX) we associate a polyhedral complex Sk(X,MX), called the skeleton which embeds in (X,MX)an along with a deformation retraction to this complex. When we work over the complex numbers with the Euclidean norm, we recover real torus fibrations; in the case of a non-Archimedean valued field, we recover affinoid torus fibrations. We use these skeleta to study log modifications of the moduli space of curves. This is joint work with Robert Crumplin.

Non-archimedean analytic fibrations and mirror symmetry

Abstract: Given X an algebraic variety over a discretely valued field K and X a divisorially log terminal model of X, one can construct the retraction Xan –> Sk(X). We endow Sk(X) with the structure of a non-archimedean analytic stack and prove that the map above is a flat map of non-archimedean analytic stacks. By explicitly describing the analytic stack structure, we investigate the analytic singularities of the fibration. We also study the tropical lifting problem in the category of non-archimedean analytic stacks.