## TGiF-Tropical Geometry in Frankfurt

## Next meeting

**24th January 2020 - Second meeting in Winter Semester 2019/20**

**Schedule**

**13:15-14:15 Karl Christ **(Ben-Gurion University)

*Title: Severi problem and tropical geometry*

### Abstract: The classical Severi problem is to show that the space of reduced and irreducible plane curves of fixed geometric genus and degree is irreducible. In case of characteristic zero, this longstanding problem was settled by Harris in 1986. In the first part of my talk I will give a brief overview of the ideas involved. Then, I will describe a tropical approach to studying degenerations of plane curves, which is the main ingredient to a new proof of irreducibility obtained in collaboration with Xiang He and Ilya Tyomkin. The main feature of the construction is that it works in positive characteristic, where the other known techniques fail.

**14:15-15:00: Coffee Break**

**15:00-16:00**** ****Oliver Lorscheid **(IMPA Rio de Janeiro/MPI Bonn)

*Title: Towards a cohomological understanding of the tropical Riemann Roch theorem*

### Abstract: In this talk, we outline a program of developing a cohomological understanding of the tropical Riemann Roch theorem and discuss the first established steps in detail. In particular, we highlight the role of the tropical hyperfield and explain why ordered blue schemes provide a satisfying framework for tropical scheme theory.

In the last part of the talk, we turn to the notion of matroid bundles, which we hope to be the right tool to set up sheaf cohomology for tropical schemes. This is based on a joint work with Matthew Baker.

**16:15-17:15 Diane Maclagan **(University of Warwick)

*Title: Connectivity of tropical varieties*

### Abstract: The structure theorem for tropical geometry states that the tropicalization of an irreducible subvariety of the algebraic torus over an algebraically closed field is the support of a pure polyhedral complex that is connected through codimension one. This means that the hypergraph whose vertices correspond to facets of the complex, and whose hyperedges correspond to the ridges, is connected. In this talk I will discuss joint work with Josephine Yu showing that this hypergraph is in fact d-connected (when the complex has no lineality space). This can be thought of as a generalization of Balinski's theorem on the d-connectivity of the edge graph of a d-polytope. A key ingredient of the proof is a toric Bertini theorem of Fuchs, Mantova, and Zannier, plus additions of Amoroso and Sombra.

## Past meetings

**31st October 2019 - First meeting in Winter Semester 2019/20**

**Schedule**

**14:00-15:00 Enrica Mazzon **(Max-Planck-Institute Bonn)

*Title: *Tropical affine manifolds in mirror symmetry

### Abstract: Mirror symmetry is a fast-moving research area at the boundary between mathematics and theoretical physics. Originated from observations in string theory, it suggests that certain geometrical objects (complex Calabi-Yau manifolds) should come in pairs, in the sense that each of them has a mirror partner and the two share interesting geometrical properties. In this talk I will introduce some notions relating mirror symmetry to tropical geometry, inspired by the work of Kontsevich-Soibelman and Gross-Siebert. In particular, I will focus on the construction of a so-called “tropical affine manifold” using methods of non-archimedean geometry, and the guiding example will be the case of K3 surfaces and some hyper-Kähler varieties. This is based on a joint work with Morgan Brown and a work in progress with Léonard Pille-Schneider.

**15:00-15:30: Coffee Break**

**15:30-16:30**** ****Christoph Goldner **(Tübingen)

*Title: Tropical mirror symmetry for ExP*^{^1}

^{^1}