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,M _{X}*) over a valued field, associates a locally ringed space (

*X,M*)

_{X}^{an}. This formalises the notion of that the Berkovich analytification of

*X*is equal to the tropicalisation of (

*X,O*). 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.

_{X}Furthermore, to each log scheme *(X,M _{X})* we associate a polyhedral complex Sk(

*X,M*), called the skeleton which embeds in (

_{X}*X,M*)

_{X}^{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 *X*^{an }–> 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.