TGIF-Tropical geometry in Frankfurt

This is an afternoon seminar series on Tropical Geometry.

The next session will be held in a hybrid format, with participants able to join both in-person and on the online platform Zoom. To participate online, 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!

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

February 3, 2023 - Second meeting in the Winter Semester 2022/23

The talks will be given in a hybrid format. If you are close-by, please join us in Frankfurt in room 711 (groß), Robert-Mayer-Str. 10, for three in-person talks. Otherwise, we're hoping to see you on Zoom. The Zoom info will be sent out to the mailing list as usual.

Schedule:

14:00-15:00 Victoria Schleis (Universität Tübingen): Linear degenerate tropical flag matroids

15:00-15:30 Break

15:30-16:30 Leonid Monin (Universität Leipzig): Polyhedral models for K-theory

16:30-16:45 Break

16:45-17:45 Navid Nabijou (University of Cambridge): Universality for tropical maps

Details:

Victoria Schleis: Linear degenerate tropical flag matroids

Abstract: Grassmannians and flag varieties are important moduli spaces in algebraic geometry. Their linear degenerations arise in representation theory as they describe quiver representations and their irreducible modules. As linear degenerations of flag varieties are difficult to analyze algebraically, we describe them in a combinatorial setting and further investigate their tropical counterparts. In this talk, I will introduce matroidal, polyhedral and tropical analoga and descriptions of linear degenerate flags and their varieties obtained in joint work with Alessio Borzì. To this end, we introduce and study morphisms of valuated matroids. Using techniques from matroid theory, polyhedral geometry and linear tropical geometry, we use the correspondences between the different descriptions to gain insight on the structure of linear degeneration. Further, we analyze the structure of linear degenerate flag varieties in all three settings, and provide some cover relations on the poset of degenerations. For small examples, we relate the observations on cover relations to the flat irreducible locus studied in representation theory.

Leonid Monin: Polyhedral models for K-theory

Abstract: One can associate a commutative, graded algebra which satisfies Poincare duality to a homogeneous polynomial f on a vector space V. One particularly interesting example of this construction is when f is the volume polynomial on a suitable space of (virtual) polytopes. In this case the algebra A_f recovers cohomology rings of toric or flag varieties.
In my talk I will explain these results and present their recent generalizations. In particular, I will explain how to associate an algebra with Gorenstein duality to any function g on a lattice L. In the case when g is the Ehrhart function on a lattice of integer (virtual) polytopes, this construction recovers K-theory of toric and full flag varieties.

Navid Nabijou: Universality for tropical maps

Abstract: I will discuss recent work concerning maps from tropical curves to orthants. A “combinatorial type" of such map is the data of an abstract graph together with slope vectors along the edges. To each such combinatorial type there is an associated moduli space, which parametrises metric enhancements of the graph compatible with the given slopes. This moduli space is a rational polyhedral cone, giving rise to an affine toric variety.Our main result shows that every rational polyhedral cone appears as the moduli space associated to some combinatorial type of tropical map. This establishes universality (also known as Murphy's law) for tropical maps. The proof is constructive and extremely concrete, as I will demonstrate. Combined with insights from logarithmic geometry, our result implies that every toric singularity appears as a virtual singularity on a moduli space of stable logarithmic maps.
This is joint work with Gabriel Corrigan and Dan Simms.

July 8, 2022 - Third meeting in the Summer Semester 2022

The talks will be given in a hybrid format. If you are close-by, please join us in Frankfurt in room 711 (groß), Robert-Mayer-Str. 10, for three in-person talks. Otherwise, we're hoping to see you on Zoom. The Zoom info will be sent out to the mailing list as usual.

Schedule:

14:00-15:00 Matilde Manzaroli (Universität Tübingen): Tropical homology over discretely valued fields

15:00-15:30 Break

15:30-16:30 Daniel Corey (Technische Universität Berlin): Initial degenerations of flag varieties

16:30-16:45 Break

16:45-17:45 Dmitry Zakharov (Central Michigan University): An analogue of Kirchhoff's theorem for the tropical Prym variety

Details:

Matilde Manzaroli: Tropical homology over discretely valued fields.

Abstract:  The talk is about a work in progress with Emiliano Ambrosi.
Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin and Ilia Zharkov proved in “Tropical homology" that for a smooth proper family of complex varieties over the punctured disk with smooth tropicalisation X the Hodge numbers of the general fiber coincide with the dimensions of the tropical homology groups of X. We explore the possibility of extending this result over more general discrete valued fields of arithmetic interest, such as R((t)) or Qp, the field of p-adique numbers. In the process of doing this, we get an action of the Galois group on the tropical homology groups and we compare this action, in certain cases, with the action defined by Tyler Foster in “Galois actions on analytifications and tropicalisation".

Daniel Corey: Initial degenerations of flag varieties

Abstract: We prove that the initial degenerations of the type-A flag variety admit closed immersions into finite inverse limits of flag matroid strata, where the diagrams are derived from matroidal subdivisions of a suitable flag matroid polytope. As an application, we prove that the initial degenerations of Fl_0(n)---the open subvariety of the complete flag variety Fl(n) consisting of flags in general position---are smooth and irreducible when n ≤ 4. We also study the Chow quotient of Fl(n) by the diagonal torus of PGL(n), and show that, for n=4, this is a log crepant resolution of its log canonical model. This is based on joint work with Jorge Alberto Olarte.

Dmitry Zakharov: An analogue of Kirchhoff's theorem for the tropical Prym variety

Abstract: The Jacobian of a finite graph is a finite abelian group, and Kirchhoff's celebrated matrix tree theorem computes the order of the Jacobian as the number of spanning trees of the graph. The Jacobian Jac(G) of a metric graph G is a real torus of dimension equal to b_1(G), and a weighted version of Kirchhoff's theorem expresses the volume of Jac(G) as a weighted sum over all spanning trees of G.

A recent paper of An, Baker, Kuperberg, and Shokrieh gives a geometric interpretation of the weighted matrix-tree theorem of a metric graph G, based on an earlier result of Mikhalkin and Zharkov. Namely, each element of Jac(G) is represented by a unique (up to translation) so-called break divisor. The type of break divisor defines a canonical cellular decomposition of Jac(G), and the individual terms in the volume formula for Jac(G) are the volumes of the cells. 

I will state and prove analogous results for the tropical Prym variety Pr(G'/G) associated to a double cover of metric graphs G'->G, as defined by Jensen, Len, and Ulirsch. The volume of Pr(G'/G) is calculated as a weighted sum over certain collections of spanning cycles on the target graph G, generalizing a similar result of Zaslavsky, Reiner and Tseng for ordinary graphs. I will then give a geometric interpretation of the volume formula in terms of a semi-canonical representability result for Prym divisors. I will discuss possible applications to the problem of resolving the Prym-Torelli map.

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

Schedule:

15:00-16:00 Dave Jensen (University of Kentucky): Brill-Noether Theory over the Hurwitz Space

16:00-16:15 Break

16:15-17:15 Kaelin Cook-Powell (Emory University): The combinatorics of the Brill-Noether Theory of general covers

Details:

Dave Jensen: Brill-Noether Theory over the Hurwitz Space

Abstract:  Brill-Noether theory is the study of line bundles on algebraic curves. A series of results in the 80's describe the varieties parameterizing line bundles with given invariants on a sufficiently general curve. More recently, several mathematicians have turned their attention to the Brill-Noether theory of general covers -- that is, curves that are general in the Hurwitz space rather than in the moduli space of curves. We will survey these recent results and, time permitting, some generalizations.

Kaelin Cook-Powell: The combinatorics of the Brill-Noether Theory of general covers

Abstract: The study of line bundles on algebraic curves has historically had deep connections with combinatorics. For example, standard young tableaux have been used to study line bundles of sufficiently general curves. Recently a variation of tableaux, known as k-uniform displacement tableaux, have been used to study line bundles of general covers -- that is curves general in the Hurwitz space. We will discuss how these displacement tableaux relate to line bundles of general covers and examine how they are used to produce new results in Brill-Noether Theory.

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

Schedule:

15:00-16:00 Ana María Botero (Universität Regensburg): Toroidal b-divisors and Monge-Ampère measures

16:00-16:30 Break

16:30-17:30 José Ignacio Burgos Gil (Instituto de Ciencias Matemáticas): Chern-Weil theory and Hilbert-Samuel theorem for semi-positive singular toroidal metrics on line bundles

Details:

Ana María Botero: Toroidal b-divisors and Monge-Ampère measures

Abstract: A b-divisor on a smooth projective algebraic variety X is an element in the projective limit of divisors, indexed over all smooth proper modifications of X, and satisfying some functoriality properties. Amongst all b-divisors, the so-called "toroidal" ones can be studied using convex-geometrical and tropical techniques. The aim of the talk is to give an overview of the theory. This joint work with Jose Burgos and Martin Sombra.

José Ignacio Burgos Gil: Chern-Weil theory and Hilbert-Samuel theorem for semi-positive singular toroidal metrics on line bundles

Abstract: In this talk I will report on joint work with A. Botero, D. Holmes and R. de Jong. Using the theory of b-divisors and non-pluripolar products we show that Chen-Weil theory and a Hilbert Samuel theorem can be extended to a wide class of singular semi-positive metrics. We apply the techniques relating semipositive metrics on line bundles to b-divisors to study the line bundle of Siegel-Jacobi forms with the Peterson metric. On the one hand we prove that the ring of Siegel-Jacobi forms of constant positive relative index is never finitely generated, and we recover a formula of Tai giving the asymptotic growth of the dimension of the spaces of Siegel-Jacobi modular forms.

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)

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

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.

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.

15:30-16:30 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

Schedule

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.