We have the great pleasure of having Kris Shaw from U.Oslo. They are a world leading expert in tropical geometry, so all the grad students that have been through my semester long bambling and rambling on the topic have a chance to hear the real deal.

Title and abstract:

Cohomologically tropical varieties

This talk asks which tropicalisations of subvarieties of the torus know the cohomology of the original variety. A motivating example are linear embeddings of complements of hyperplane arrangements.
We can prove that the tropicalisation knows the cohomology of the variety in a strong sense if and only if it satisfies local tropical Poincaré duality and the original variety is so-called “wunderschön”. Following the work of Itenberg, Katzarkov, Mikhalkin, and Zharkov, we can obtain information about the mixed Hodge structure of a family of varieties from its tropicalisation when it locally satisfies the two conditions above.

This is joint work with Edvard Aksnes, Omid Amini, and Matthieu Piquerez