## TGiF-Tropical Geometry in Frankfurt

## Next meetings on Zoom

**TGIZ-Tropical geometry in Zoom**

*This is an afternoon seminar series on Tropical Geometry.*

*The next sessions will be held on the online platform Zoom. Please register by sending an email to one of the organisers by the day before the next meeting. You will then receive the link to the meeting on the day of the meeting. Those in the SGA mailing list need not register.*

*TGIZ was paused during Fall 2021 and replaced by a series of Workshops on Non-Archimedean and tropical geometry. We have started again in January 2022!*

*TGIZ was paused during Fall 2021 and replaced by a series of Workshops on Non-Archimedean and tropical geometry. We have started again in January 2022!*

*Videos of some of the past talks are now available via our new ***youtube channel**.

**youtube channel**.

**May 13, 2022 - First meeting in the Summer Semester 2022**

**15:00-17:30 Ana María Botero and José Ignacio Burgos Gil**

**June 10, 2020 - Second meeting in the Summer Semester 2022**

**15:00-17:30 Dave Jensen and Kaelin Cook-Powell**

## Past online meetings

**February 18th, 2022 - Second meeting in the Winter Semester 2021/22**

**Schedule **

**14:00-15:00 Johannes Rau (Universidad de los Andes): ***Patchworks of real algebraic varieties in higher codimension*

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

**15:15-16:15 Siddarth Kannan (Brown University): ***Cut-and-paste invariants of moduli spaces of relative stable maps to P^1*

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

**16:30-17:30 Rohini Ramadas (University of Warwick): ***The S_n action on the homology groups of M_{0,n}-bar*

**Details:**

*Johannes Rau: *Patchworks of real algebraic varieties in higher codimension

### Abstract: I will present a combinatorial setup, based on smooth tropical varieties and real phase structures, which after "unfolding" produces a certain class of PL-manifolds (called patchworks). We have two motivations in mind: Firstly, in the spirit of Viro's combinatorial patchwoking for hypersurfaces, these patchworks can be used to describe the topology of real algebraic varieties close to the tropical limit. Secondly, even if not "realisable" by real algebraic varieties, real phase structures provide a geometric framework for combinatorial structures such as oriented matroids. Joint work with Arthur Renaudineau and Kris Shaw.

*Siddarth Kannan: Cut-and-paste invariants of moduli spaces of relative stable maps to P^1*

### Abstract: I will discuss ongoing work studying moduli spaces of genus zero stable maps to P^1, with fixed ramification profiles over 0 and infinity. I will describe a chamber decomposition of the space of ramification data such that the Grothendieck class of the moduli space is constant on the chambers. Finally, for the sequence of ramification data corresponding to maximal ramification over 0 and no ramification over infinity, I will describe a recursive algorithm to compute the generating function for Euler characteristics of these spaces.

*Rohini Ramadas: The S_n action on the homology groups of M_{0,n}-bar*

### Abstract: The symmetric group on n letters acts on M_{0,n}-bar, and thus on its (co-)homology groups. The induced actions on (co-)homology have been studied by, eg., Getzler, Bergstrom-Minabe, Castravet-Tevelev. We ask: does H_{2k}(M_{0,n}-bar) admit an equivariant basis, i.e. one that is permuted by S_n? We describe progress towards answering this question. This talk includes joint work with Rob Silversmith.

**January 21st, 2022 - First meeting in the Winter Semester 2021/22**

**Schedule **

**14:00-15:00 Mima Stanojkovski (RWTH Aachen): ***Orders and polytropes: matrices from valuations*

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

**15:15-16:15 Ilya Tyomkin (Ben Gurion University): ***Applications of tropical geometry to irreducibility problems in algebraic geometry*

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

**16:30-17:30 Harry Richman (University of Washington): ***Uniform bounds for torsion packets on tropical curves*

**Details:**

*Mima Stanojkovski: Orders and polytropes: matrices from valuations*

### Abstract: Let K be a discretely valued field with ring of integers R. To a d-by-d matrix M with integral coefficients one can associate an R-module, in K^{d x d}, and a polytope, in the Euclidean space of dimension d-1. We will look at the interplay between these two objects, from the point of view of tropical geometry and building on work of Plesken and Zassenhaus. This is joint work with Y. El Maazouz, M. A. Hahn, G. Nebe, and B. Sturmfels.

*Ilya Tyomkin: Applications of tropical geometry to irreducibility problems in algebraic geometry*

### Abstract: In my talk, I will discuss a novel tropical approach to classical irreducibility problems of Severi varieties and of Hurwitz schemes. I will explain how to prove such irreducibility results by investigating the properties of tropicalizations of one-parameter families of curves and of the induced maps to the tropical moduli space of parametrized tropical curves. The talk is based on a series of joint works with Karl Christ and Xiang He.

*Harry Richman: Uniform bounds for torsion packets on tropical curves*

### Abstract: Say two points x, y on an algebraic curve are in the same torsion packet if [x - y] is a torsion element of the Jacobian. In genus 0 and 1, torsion packets have infinitely many points. In higher genus, a theorem of Raynaud states that all torsion packets are finite. It was long conjectured, and only recently proven*, that the size of a torsion packet is bounded uniformly in terms of the genus of the underlying curve. We study the tropical analogue of this construction for a metric graph. On a higher genus metric graph, torsion packets are not always finite, but they are finite under an additional "genericity" assumption on the edge lengths. Under this genericity assumption, the torsion packets satisfy a uniform bound in terms of the genus of the underlying graph. (*by Kuehne and Looper-Silverman-Wilmes in 2021)

**June 25th, 2021 - Third meeting in the Summer Semester 2021**

**Schedule **

**14:00-15:00 Hülya Argüz (Université de Versailles)**

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

**15:15-16:15 Stefano Mereta (Swansea University)**

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

**16:30-17:30 Eric Katz (Ohio State University)**

**Details:**

*Hülya Argüz: Tropical enumeration of real log curves in toric varieties and log Welschinger invariants*

### Abstract: We give a new proof of a central theorem in real enumerative geometry: the Mikhalkin correspondence theorem for Welschinger invariants. The proof goes through totally different techniques as the original proof of Mikhalkin and is an adaptation to the real setting of the approach of Nishinou-Siebert to the complex correspondence theorem. It uses log-geometry as a central tool. We will discuss how this reinterpretation in terms of log-geometry may lead to new developments, as for example a real version of mirror symmetry. This is joint work with Pierrick Bousseau.

*Stefano Mereta: Tropical differential equations*

### Abstract: In 2015 Dimitri Grigoriev introduced a way to tropicalize differential equation with coefficients in a power series ring and defined what a solution for such a tropicalized equation should be. In 2016 Aroca, Garay and Toghani proved a fundamental theorem analogue to the fundamental theorem of tropical geometry for power series over a trivially valued field. In this talk I will introduce the basic ideas moving then towards a functor of points approach to the subject by means of the recently developed tropical scheme theory, as introduced by Giansiracusa and Giansiracusa, looking at solutions to such equations as morphisms between so-called pairs. I will also give a generalisation to power series ring with non-trivially valued coefficients and state a colimit theorem along the lines of Payne's inverse limit theorem.

*Eric Katz: Combinatorial and p-adic iterated integrals*

### Abstract: The classical operations of algebraic geometry often have combinatorial analogues. We will discuss the combinatorial analogue of Chen’s iterated integrals. These have a richer, non-abelian structure than classical integrals. We will describe the tropical analogue of the unipotent Torelli theorem of Hain and Pulte and make connections between iterated integrals and monodromy with applications to p-adic integration.

**May 28th, 2021 - Second meeting in the Summer Semester 2021**

**Schedule **

**14:00-15:00 Margarida Melo (Roma Tre University)**

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

**15:15-16:15 Baldur Sigurðsson (UNAM Cuernavaca)**

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

**16:30-17:30 Jenia Tevelev (UMass Amherst)**

**Details**

*Margarida Melo: On the top weight cohomology of the moduli space of abelian varieties*

### The moduli space of abelian varieties Ag admits well behaved toroidal compactifications whose dual complex can be given a tropical interpretation. Therefore, one can use the techniques recently developed by Chan-Galatius-Payne in order to understand part of the topology of Ag via tropical geometry. In this talk, which is based in joint work with Madeleine Brandt, Juliette Bruce, Melody Chan, Gwyneth Moreland and Corey Wolfe, I will explain how to use this setup, and in particular computations in the perfect cone compactification of Ag, in order to describe its top weight cohomology for g up to 7.

*Baldur Sigurðsson: Local tropical Cartier divisors and the multiplicity*

### We consider the group of local tropical cycles in the local tropicalization of the local analytic ring of a toric variety, in particular, Cartier divisors defined by a function germ. We see a formula for the multiplicity, a result which is motivated by a classical theorem of Wagreich for normal surface singularities.

*Jenia Tevelev: Compactifications of moduli of points and lines in the (tropical) plane*

### Projective duality identifies moduli spaces of points and lines in the projective plane. The latter space admits Kapranov's Chow quotient compactification, studied also by Lafforgue, Hacking-Keel-Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable surfaces: it carries a family of reducible degenerations of the projective plane with "broken lines". From the tropical perspective, these degenerations are encoded in matroid decompositions and tropical planes and their moduli space in the Dressian and the tropical Grasmannian. In 1991, Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing reducible degenerations of the projective plane with n smooth points. In a joint paper with Luca Schaffler, we investigate the extension of projective duality to degenerations, answering a question of Kapranov.

**April 30th, 2021 - First meeting in the Summer Semester 2021**

**Schedule **

**14:00-15:00 Felipe Rincon (Queen Mary University)**

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

**15:15-16:15 Jeremy Usatine (Brown University)**

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

**16:30-17:30 Shiyue Li (Brown University)**

**Details**

*Felipe Rincon: Tropical Ideals*

### Tropical ideals are ideals in the tropical polynomial semiring in which any bounded-degree piece is “matroidal”. They were conceived as a sensible class of objects for developing algebraic foundations in tropical geometry. In this talk I will introduce and motivate the notion of tropical ideals, and I will discuss work studying some of their main properties and their possible associated varieties.

*Jeremy Usatine: Stringy invariants and toric Artin stacks*

### Stringy Hodge numbers are certain generalizations, to the singular setting, of Hodge numbers. Unlike usual Hodge numbers, stringy Hodge numbers are not defined as dimensions of cohomology groups. Nonetheless, an open conjecture of Batyrev's predicts that stringy Hodge numbers are nonnegative. In the special case of varieties with only quotient singularities, Yasuda proved Batyrev's conjecture by showing that the stringy Hodge numbers are given by orbifold cohomology. For more general singularities, a similar cohomological interpretation remains elusive. I will discuss a conjectural framework, proven in the toric case, that relates stringy Hodge numbers to motivic integration for Artin stacks, and I will explain how this framework applies to the search for a cohomological interpretation for stringy Hodge numbers. This talk is based on joint work with Matthew Satriano.

*Shiyue Li: Topology of tropical moduli spaces of weighted stable curves in higher genus*

### The space of tropical weighted curves of genus g and volume 1 is the dual complex of the divisor of singular curves in Hassett’s moduli space of weighted stable genus g curves. One can derive plenty of topological properties of the Hassett spaces by studying the topology of these dual complexes. In this talk, we show that the spaces of tropical weighted curves of genus g and volume 1 are simply-connected for all genus greater than zero and all rational weights, under the framework of symmetric Delta-complexes and via a result by Allcock-Corey-Payne 19. We also calculate the Euler characteristics of these spaces and the top weight Euler characteristics of the classical Hassett spaces in terms of the combinatorics of the weights. I will also discuss some work in progress on a geometric group approach to simple connectivity of these spaces. This is joint work with Siddarth Kannan, Stefano Serpente, and Claudia Yun.

**March 12th, 2021 - Fourth meeting in the Winter Semester 2020/21**

**Schedule **

**14:00-15:00 Anthea Monod (Imperial College)**

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

**15:15-16:15 Claudia He Yun (Brown University)**

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

**16:30-17:30 Daniel Corey (University of Wisconsin)**

**Details**

*Anthea Monod: Tropical geometry of phylogenetic tree spaces*

### Abstract: The Billera-Holmes-Vogtmann (BHV) space is a well-studied moduli space of phylogenetic trees that appears in many scientific disciplines, including computational biology, computer vision, combinatorics, and category theory. Speyer and Sturmfels identify a homeomorphism between BHV space and a version of the Grassmannian using tropical geometry, endowing the space of phylogenetic trees with a tropical structure, which turns out to be advantageous for computational studies. In this talk, I will present the coincidence between BHV space and the tropical Grassmannian. I will then give an overview of some recent work I have done that studies the tropical Grassmannian as a metric space and the practical implications of these results on probabilistic and statistical studies on real datasets of phylogenetic trees.

*Claudia Yun: The S_n-equivariant rational homology of the tropical moduli spaces \Delta_{2,n}*

### Abstract: The tropical moduli space $\Delta_{g,n}$ is a topological space that parametrizes isomorphism classes of $n$-marked stable tropical curves of genus $g$ with total volume 1. Its reduced rational homology has a natural structure of $S_n$-representations induced by permuting markings. In this talk, we focus on $\Delta_{2,n}$ and compute the characters of these $S_n$-representations for $n$ up to 8. We use the fact that $\Delta_{2,n}$ is a symmetric $\Delta$-complex, a concept introduced by Chan, Glatius, and Payne. The computation is done in SageMath.

### Daniel Corey: The Ceresa class: tropical, topological and algebraic

### Abstract: The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve. It is homologically trivial but not algebraically equivalent to zero for a very general curve. In this sense, it is one of the simplest algebraic cycles that goes ``beyond homology.'' The image of the Ceresa cycle under a certain cycle class map produces a class in étale homology called the Ceresa class. We define the Ceresa class for a tropical curve and for a product of commuting Dehn twists on a topological surface. We relate these to the Ceresa class of a smooth algebraic curve over C((t)). Our main result is that the Ceresa class in each of these settings is torsion. Nevertheless, this class is readily computable, frequently nonzero, and implies nontriviality of the Ceresa cycle when nonzero. This is joint work with Jordan Ellenberg and Wanlin Li.

**February 19th, 2021 - Third meeting in the Winter Semester 2020/21**

**Schedule **

**14:00-15:00 John Christian Ottem (University of Oslo)**

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

**15:15-16:15 Marco Pacini (Universidade Federal Fluminense)**

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

**16:30-17:30 Laura Escobar (Washington University in St. Louis)**

**Details**

*John Christian Ottem: Tropical degenerations and stable rationality*

### I will explain how tropical degenerations and birational specialization techniques can be used in rationality problems. In particular, I will apply these techniques to study quartic fivefolds and complete intersections of a quadric and a cubic in P^6. This is joint work with Johannes Nicaise.

*Marco Pacini: A universal tropical Jacobian over the moduli space of tropical curves. *

### Abstract: We introduce polystable divisors on a tropical curve, which are the tropical analogue of polystable torsion-free rank-1 sheaves on a nodal curve. We show how to construct a universal tropical Jacobian by means of polystable divisors on tropical curves. This space can be seen as a tropical counterpart of Caporaso's universal Picard scheme. This is a joint work with Abreu, Andria, and Taboada.

### Laura Escobar: Wall-crossing for Newton-Okounkov bodies

### Abstract: A Newton-Okounkov body is a convex set associated to a projective variety, equipped with a valuation. These bodies generalize the theory of Newton polytopes. Work of Kaveh-Manon gives an explicit link between tropical geometry and Newton-Okounkov bodies. In joint work with Megumi Harada we use this link to describe a wall-crossing phenomenon for Newton-Okounkov bodies.

**January 22nd, 2021 - Second meeting in the Winter Semester 2020/21**

**Schedule **

**14:00-15:00 Alheydis Geiger (Universität Tübingen)**

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

**15:15-16:15 Matt Baker (Georgia Institute of Technology) **

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

**16:30-17:30 Daniel Litt (University of Georgia)**

**Details**

* Alheydis Geiger: Deformations of bitangent classes of tropical quartic curves *

*Abstract:* Over an algebraically closed field a smooth quartic curve has 28 bitangent lines. Plücker proved that over the real numbers we have either 4, 8, 16 or 28 real bitangents to a real quartic curve. A tropical smooth quartic curve has exactly 7 bitangent classes which each lift either 0 or 4 times over the real numbers. The shapes of these bitangent classes have been classified by Markwig and Cueto in 2020, who also determined their real lifting conditions. However, for a fixed unimodular triangulation different choices of coefficients imply different edge lengths of the quartic and these can change the shape of the 7 bitangent classes and might therefore influence their real lifting conditions. In order to prove Plückers Theorem about the number of real bitangents tropically, we have to study these deformations of the bitangent shapes. In a joint work with Marta Panizzut we develope a polymake extension, which computes the tropical bitangents. For this we determine two refinements of the secondary fan: one for which the bitangent shapes in each cone stay constant and one for which the lifting conditions in each cone stay constant. This is still work in progress, but there will be a small software demonstration.

*Matt Baker: Pastures, Polynomials, and Matroids*

*Abstract:* A pasture is, roughly speaking, a field in which addition is allowed to be both multivalued and partially undefined. Pastures are natural objects from the point of view of F_1 geometry and Lorscheid’s theory of ordered blueprints. I will describe a theorem about univariate polynomials over pastures which simultaneously generalizes Descartes’ Rule of Signs and the theory of NewtonPolygons. Conjecturally, there should be a similar picture for several polynomials in several variables generalizing tropical intersection theory. I will also describe a novel approach to the theory of matroid representations which revolves around a canonical universal pasture called the “foundation” that one can attach to any matroid. This is joint work with Oliver Lorscheid.

*Daniel Litt: The tropical section conjecture*

*Abstract:* Grothendieck's section conjecture predicts that for a curve X of genus at least 2 over an arithmetically interesting field (say, a number field or p-adic field), the étale fundamental group of X encodes all the information about rational points on X. In this talk I will formulate a tropical analogue of the section conjecture and explain how to use methods from low-dimensional topology and moduli theory to prove many cases of it. As a byproduct, I'll construct many examples of curves for which the section conjecture is true, in interesting ways. For example, I will explain how to prove the section conjecture for the generic curve, and for the generic curve with a rational divisor class, as well as how to construct curves over p-adic fields which satisfy the section conjecture for geometric reasons. This is joint work with Wanlin Li, Nick Salter, and Padma Srinivasan.

**4. Dec. 2020 - First meeting in Winter Semester 2020/21**

**Schedule **

**14:00-15:00 Xin Fang (University of Cologne)**

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

**15:15-16:15 Man-Wai Cheung (Harvard University)**

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

**16:30-17:30 Lara Bossinger (UNAM Oaxaca)**

**Abstracts**

* Xin Fang: Tropical flag varieties - a Lie theoretic approach*

### In this talk I will explain how to use Lie theory to describe the facets of a maximal prime cone in a type A tropical complete flag variety. The face lattice of this cone encodes degeneration structures in Lie algebra, quiver Grassmannians and module categories of quivers. This talk bases on different joint works with (subsets of) G. Cerulli-Irelli, E. Feigin, G. Fourier, M. Gorsky, P. Littelmann, I. Makhlin and M. Reineke, as well as some work in progress.

*Man-Wai Cheung: Polytopes, wall crossings, and cluster varieties*

### Cluster varieties are log Calabi-Yau varieties which are a union of algebraic tori glued by birational "mutation" maps. Partial compactifications of the varieties, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties. However, it is not clear from the definitions how to characterize the polytopes giving compactifications of cluster varieties. We will show how to describe the compactifications easily by broken line convexity. As an application, we will see the non-integral vertex in the Newton Okounkov body of Gr(3,6) comes from broken line convexity. Further, we will also see certain positive polytopes will give us hints about the Batyrev mirror in the cluster setting. The mutations of the polytopes will be related to the almost toric fibration from the symplectic point of view. Finally, we can see how to extend the idea of gluing of tori in Floer theory which then ended up with the Family Floer Mirror for the del Pezzo surfaces of degree 5 and 6. The talk will be based on a series of joint works with Bossinger, Lin, Magee, Najera-Chavez, and Vianna.

*Lara Bossinger: Tropical geometry of Grassmannians and their cluster structure*

### Abstract: The Grassmannain, or more precisely its homogeneous coordinate ring with respect to the Plücker embedding, was found to be a cluster algebra by Scott in the early years of cluster theory. Since then, this cluster structure was studied from many different perspectives by a number of mathematicians. As the whole subject of cluster algebras broadly speaking divides into two main perspectives, algebraic and geometric, so do the results regarding Grassmannian. Geometrically, the Grassmannian contains two open subschemes that are dual cluster varieties.

### Interestingly, we can find tropical geometry in both directions: from the algebraic point of view, we discover relations between maximal cones in the tropicalization of the defining ideal (what Speyer and Sturmfels call the tropical Grassmannian) and seeds of the cluster algebra. From the geometric point of view, due to work of Fock--Goncharov followed by work of Gross--Hacking--Keel--Kontsevich we know that the scheme theoretic tropical points of the cluster varieties parametrize functions on the Grassmannian.

### In this talk I aim to explain the interaction of tropical geometry with the cluster structure for the Grassmannian from the algebraic and the geometric point of view.

**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.

**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}