## TGiF-Tropical Geometry in Frankfurt

## Next meeting on Zoom

### TGIZ-Tropical geometry in Zoom

**Videos of the past talks are available via ****this link.**

**Videos of the past talks are available via**

**this link.**

**The next seminar will be held on the online platform Zoom. It is possible to register sending an email to one of the organisers by the 25 of June and then receive the link and password to the meeting.**

**The next seminar will be held on the online platform Zoom. It is possible to register sending an email to one of the organisers by the 25 of June and then receive the link and password to the meeting.**

**Those in the SGA mailing list need not to register.**

**Those in the SGA mailing list need not to register.**

**26th June 2020 - Third meeting in Summer Semester 2020**

**Schedule**

**14:00-15:00: Mark Gross (University of Cambridge)**

*Gluing log Gromov-Witten invariants*

### 15:00-15:15: Break

**15:15-16:15: Luca Battistella (Universität Heidelberg) **

*A smooth compactification of genus two curves in projective space*

### 16:15-16:30: Break

**16:30-17:30: Kalina Mincheva (Yale University)**

*Prime tropical ideals*

**Abstracts**

* Gluing log Gromov-Witten invariants(Mark Gross)*

### I will give a progress report on joint work with Abramovich, Chen and Siebert aiming to understand gluing formulae for log Gromov-Witten invariants, generalizing the Li/Ruan and Jun Li gluing formulas for relative Gromov-Witten invariants.

*A smooth compactification of genus two curves in projective space (Luca Battistella)*

### Questions of enumerative geometry can often be translated into problems of intersection theory on a compact moduli space of curves in projective space. Kontsevich's stable maps work extraordinarily well when the curves are rational, but in higher genus the burden of degenerate contributions is heavily felt, as the moduli space acquires several boundary components. The closure of the locus of maps with smooth source curve is interesting but troublesome, for its functor of points interpretation is most often unclear; on the other hand, after the work of Li--Vakil--Zinger and Ranganathan--Santos-Parker--Wise in genus one, points in the boundary correspond to maps that admit a nice factorisation through some curve with Gorenstein singularities (morally, contracting any higher genus subcurve on which the map is constant). The question becomes how to construct such a universal family of Gorenstein curves. In joint work with F. Carocci, we construct one such family in genus two over a logarithmic modification of the space of admissible covers. I will focus on how tropical geometry determines this logarithmic modification via tropical canonical divisors.

*Prime tropical ideals. (Kalina Mincheva)*

### In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences to conclude that the variety of a non-zero prime (tropical) ideal is either empty or consists of a single point.

## Past online meetings

**29th May 2020 - Second meeting in Summer Semester 2020**

**Schedule**

### 14:00-15:00: Ben Smith (University of Manchester) (cancelled)

*Faces of tropical polyhedra.*

### 15:00-15:15: Break

**15:15-16:15: Yue Ren (University of Swansea) **

*Tropical varieties of neural networks *

### 16:15-16:30: Break

**16:30-17:30: Hannah ****Markwig** (Eberhard Karls Universität Tübingen)

**Markwig**(Eberhard Karls Universität Tübingen)

*The combinatorics and real lifting of tropical bitangents to plane quartics *

**Abstracts**

*Faces of tropical polyhedra (Ben Smith)*

### Tropical polyhedra are tropicalizations of ordinary polyhedra, and have found applications in many areas of pure and applied mathematics. While they have many nice combinatorial properties, the notion of a "face" of a tropical polyhedron has been difficult to define. In this talk, we shall examine the obstacles that arise when considering faces of tropical polyhedra. We also offer a possible solution by defining faces for a special class of tropical polyhedra arising as tropicalisations of blocking polyhedra. We then show how this face structure may be extended to all tropical polyhedra. This is joint work with Georg Loho.

*Tropical varieties of neural networks (Yue Ren)*

In this talk, we introduce tropical varieties arising from neural

networks with piecewise linear activation function. We show how

Stiefel tropical linear spaces correspond to special maxout networks

and compare Speyer's f-Vector Theorem with existing results in machine

learning on their complexity. We briefly touch upon the notion of

Vapnik-Chervonenkis dimension of neural networks and conclude with

some open questions in tropical geometry. This is joint work with

Kathryn Heal (Harvard), Guido Montufar (UCLA + MPI MiS), and Leon Zhang (UC Berkeley).

*The combinatorics and real lifting of tropical bitangents to plane quartics (Hannah Markwig)*

A plane quartic has 28 bitangents. A tropical plane quartic may have

infinitely many bitangents, but there is a natural equivalence relation

for which we obtain precisely 7 bitangent classes. If a tropical quartic

is Trop(V(q)) for a polynomial q in K[x,y] (where K is the field of

complex Puiseux series), it is a natural question where in the 7

bitangent classes the tropicalizations of the 28 bitangents of V(q) are,

or, put differently, which member of the tropical bitangent classes

lifts to a bitangent of V(q), and with what multiplicity. It is not

surprising that each bitangent class has 4 lifts. If q is defined over

the reals, V(q) can have 4, 8, 16 or 28 real bitangents. We show that

each tropical bitangent class has either 0 or 4 real lifts - that is,

either all complex solutions are real, or none. We also discuss further

questions concerning tropical tangents, their combinatorics and their

real lifts. This talk is based on joint work with Yoav Len, and with

Maria Angelica Cueto.

**24th April 2020 - First meeting in Summer Semester 2020**

**Schedule**

**14:00-15:00: Marta Panizzut (Universität Osnabrück)**

*Tropical cubic surfaces and their lines. *

### 15:00-15:15: Break

**15:15-16:15: Jan Draisma (Universität Bern)**

*Catalan-many morphisms to trees-Part I*

### 16:15-16:30: Break

**16:30-17:30: Alejandro Vargas (Universität Bern)**

*Catalan-many morphisms to trees-Part II*

**Abstracts**

*Tropical cubic surfaces and their lines (Marta Panizzut)*

In this talk we investigate different models to study tropical cubic surfaces and their 27 lines.

First we look at smooth tropical cubic surfaces and the combinatorics of their lines in tropical 3-dimensional torus.

We then focus on the tropicalization of the moduli space of del Pezzo surfaces of degree three as in the work of Ren, Shaw and Sturmfels.

Finally we introduce an octanomial model for cubic surfaces. This new normal form is well suited for p-adic geometry,

as it reveals the intrinsic del Pezzo combinatorics of the 27 lines in the tropicalization.

### The talk is based on joint work with Micheal Joswig, Emre Sertöz and Bernd Sturmfels.

*Catalan-many morphisms to trees (Jan Draisma-Alejandro Vargas)*

### Abstract: We report on a several-year project, recently completed, to find a purely combinatorial proof for the result that a genus-g metric graph admits a tropical morphism of genus 1+\lceil g/2 \rceil to a metric tree. The proofs of this result so far have been via specialisation lemmas due to Baker and Caporaso that tropicalize the analogous fact from algebraic geometry.

### We also give a preview on the forthcoming sequel where we count the number of such tropical morphisms in the even genus case and, under a suitable notion of multiplicity, obtain a Catalan number.

### Jan Draisma: introduction to theorem, relation with classical theory

### Alejandro Vargas: key ideas of proof

## Past meetings

**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.

**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}