I will speak about my work with Helge Ruddat on how to construct explicitly log structures and morphisms. I will also discuss some motivation. I will try to stay informal and assume no prior knowledge of log structures.
Sometimes artists become better known for their lifestyle or fashion tastes than the work without which we wouldn't know or care about them - Van Gogh, Bowie, even Taylor Swift. Jim Morrison of the Doors fits that bill, in his case for his death and grave as much as his life. But he and the band wrote some songs. That's where it all starts and ends.
Modern approaches to infinitesimal deformations of algebro-geometric objects (like varieties) use the setting of formal moduli problems, from derived geometry. It allows to prove that all kinds of deformations are governed by a tangent complex equipped with a derived Lie algebra structure. I will use this framework to study equivariant deformations of varieties with respect to the action of an algebraic group. Then, I will explain how this theory of equivariant deformations allows us to prove a dichotomous behaviour for almost all varieties that are homogeneous under a reductive group : either they deform to characteristic 0, or they admit no deformation to any ring of characteristic greater than p.
I work in the field of geometric group theory. This is a pretty broad heading, but for me it means that the goal is to understand infinite groups, and the strategy is to get them to act on nice metric spaces that we know a lot about.
