TGiF-Tropical Geometry in Frankfurt

Next meeting on Zoom

TGIZ-Tropical geometry in Zoom

This is an afternoon seminar series on Tropical Geometry.

The next session will be held on the online platform Zoom. Please register by sending an email to one of the organisers by the Jan. 21st. You will then receive the link to the meeting on Jan. 22nd.

Those in the SGA mailing list need not to register.

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

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


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)


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.

Future Meetings

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


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)


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: TBA

Abstract: TBA

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


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)



Past online meetings

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


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)


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


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



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


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)

The combinatorics and real lifting of tropical bitangents to plane quartics 


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


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


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


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


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

Abstract: We recall some results of tropical mirror symmetry that relate the generating series of tropical Gromov-Witten invariants of an elliptic curve E to sums of Feynman integrals. After that, we present an approach to tropical mirror symmetry in case of ExP^1. The approach is based on the floor decomposition of tropical curves which is a degeneration technique that allows us to apply the results of the elliptic curve case. The new results are joint work with Janko Böhm and Hannah Markwig.

16:45-17:45 Sam Payne (University of Texas, Austin)

Title: Local h-vectors

Abstract: TBA

5th July 2019 - Second meeting in Summer Semester


13:15-14:15 Madeline Brandt (University of California at Berkeley)

Title: Matroids and their Dressians

Abstract: In this talk we will explore Dressians of matroids. Dressians have many lives: they parametrize tropical linear spaces, their points induce regular matroid subdivisions of the matroid polytope, they parametrize valuations of a given matroid, and they are a tropical prevariety formed from certain Plücker equations. We show that initial matroids correspond to cells in regular matroid subdivisions of matroid polytopes, and we characterize matroids that do not admit any proper matroid subdivisions. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. If time permits, we will also discuss an ongoing project extending these ideas to flag matroids.

14:45-15:45 Dhruv Ranganathan (University of Cambridge)

Tropical curves, stable maps, and singularities in genus one

In the early days of tropical geometry, Speyer identified an extremely subtle combinatorial condition that distinguished tropical elliptic space curves from arbitrary balanced genus one graphs. Just before this, Vakil and Zinger gave a very explicit desingularization of the moduli space of elliptic curves in projective space, with remarkable applications. Just after this, Smyth constructed new compactifications of moduli spaces of pointed elliptic curves, using worse-than-nodal singularities, as part of the Hasset-Keel program. A decade on, we understand these three results as part of a single story involving logarithmic structures and their tropicalizations. I will discuss this picture and how the unified framework extends all three results. This is joint work with Keli Santos-Parker and Jonathan Wise. 

16:00-17:00 Yoav Len (Georgia Institute of Technology)

Algebraic and Tropical Prym varieties

My talk will revolve around combinatorial aspects of Abelian varieties. I will focus on Pryms, a class of Abelian vari- eties that occurs in the presence of double covers, and have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures. I will explain how problems concern- ing Pryms may be reduced, via tropical geometry, to problems on metric graphs. As a consequence, we obtain new results con- cerning the geometry of special algebraic curves, and bounds on dimensions of certain Brill–Noether loci. This is joint work with Martin Ulirsch.

7th June 2019 - First meeting in Summer Semester

Margarida Melo (Università degli studi Roma Tre):

Combinatorics and moduli of line bundles on stable curves.

 The moduli space of line bundles on smooth curves of given genus, the so called universal Jacobian, has a number of different compactifications over the moduli space of stable curves. These compactificatons have very interesting combinatorial properties, which can be used to describe their geometry. In the talk I will explain different features and applications of these interesting objects, focusing on properties which have a natural tropical counterpart.

Farbod Shokrieh (University of Copenhagen): 

Heights and moments of abelian varieties

We give a formula which, for a principally polarized abelian variety $(A,\lambda)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the N\'eron-Tate height of $(A,\lambda)$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. We also discuss the case of Jacobians in some detail, where graphs and electrical networks will play a role.

(Based on joint works with Robin de Jong.)

Philipp Jell (Universität Regensburg):

The tropical Hodge conjecture for divisors 

The Hodge conjecture is one of the big open questions in algebraic geometry. Mikhalkin and Zharkov formulated a tropical analogue of this conjecture. In joint work with Johannes Rau and Kristin Shaw, we established this conjecture for divisors. I will introduce the notions that are necessary to state the tropical Hodge conjecture and then sketch the proof and further directions of research.