# A Tropical Panorama

## Abstracts for the talks

**Gavril Farkas ***Humboldt-Universität zu Berlin*

Time: Monday, July 09, 2018, 11:00

**Green's Conjecture: a new approach via Chow forms of Grassmannians**

I will discuss and prove an algebraic statement concerning the vanishing of the Koszul modules associated to any subspace inside the second exterior product of a complex vector space. This statement turns out to be equivalent to Mark Green's Conjecture on syzygies of canonical curves of genus g, which gives an alternative to Voisin's proof (in characteristic zero) and a first proof of Green's conjecture in characteristic p>(g+3)/2. I will also present topological applications of this result, focusing on an explicit description of the Cayley-Chow form of the Grassmannian of lines. This is joint work with M. Aprodu, S. Papadima, C. Raicu and J. Weyman.

**Tyler Foster ***Max-Planck-Institut für Mathematik Bonn*

Time: Monday, July 09, 2018, 13:00

**Mirrors of quotients and quotients of mirrors**

Ideas from tropical geometry play an important role in Gross, Hacking, and Keel's construction of mirrors for log-Calabi-Yau varieties. In this talk, I'll present work in progress with Colin Diemer in which we study the structure of GHK-mirrors under the action of a finite group. When a log-CY variety X comes equipped with the left-action of a finite group G, its GHK-mirror family X* acquires a dual right-action by the same group. Using this dual action, we show that in many cases, the mirror family of the left-quotient G\X can be naturally realized as the pullback of a stratum inside the right-quotient X*/G of the original mirror family X* (up to a uniform re-scaling of the relevant instanton corrections). More concisely: the mirror family (G\X)* is "almost" a canonically defined piece of the quotient family X*/G. In invariant theoretic terms: the broken line counts that give the structure constants for the coordinate ring of the mirror (G\X)* already appear as structure constants in the ring of G-invariants on the mirror X* (again up to uniform re-scaling).

**Jeffrey Giansiracusa ***Swansea University*

Time: Monday, July 09, 2018, 15:50

**Exploring tropical ideals**

The scheme-theoretic approach to tropical geometry has motivated the study of "tropical ideals", which are sets of tropical polynomials that form an ideal whose graded pieces are tropical linear spaces. There are realizable tropical ideals, meaning that they are formed by tropicalizing classical ideals as linear spaces, and there are non-realizable tropical ideals. Three interesting questions are:

1) What invariants of a classical ideal are encoded in its associated tropical ideal?

2) How does the tropicalization of an ideal change as the ideal changes (moving within the Hilbert scheme)?

3) How can one construct non-realizable tropical ideals?

In this talk I will discuss examples, progress and open questions on each of these questions.

**Alexander Guterman ***Lomonosov Moscow State University*

Time: Tuesday, July 10, 2018, 15:00

**Tropical matrix invariants and maps preserving them**

We discuss different matrix invariants and relations which are important in tropical linear algebra. In particular we characterize linear maps preserving these invariants and relations.

**Sara Lamboglia ***Goethe University Frankfurt*

Time: Tuesday, July 10, 2018, 15:50

**Tropical Fano schemes**

The classical Fano scheme F_{d}(X) of a variety X parametrises d-dimensional linear spaces contained in X. In this talk I am going to deﬁne the tropical analogue of the Fano scheme F_{d}(tropX) and I will show its relation with the tropicalization tropF_{d}(X) of the classical Fano scheme. In particular I will focus on the tropical Fano schemes of tropicalized linear spaces and tropicalized toric varieties.

**Georg Loho ***Ecole polytechnique fédérale de Lausanne*

Time: Tuesday, July 10, 2018, 13:50

**Monomial tropical cones for multicriteria optimization**

We introduce a special class of tropical cones which we call 'monomial tropical cones'. They arise as a helpful tool in the description of discrete multicriteria optimization problems. After an introduction to tropical convexity with an emphasis on these particular tropical cones, we explain the algorithmic implications. It leads to an efficient algorithm for computing all nondominated points of a discrete multicriteria optimization problem based on tropical convex hull computation.

We finish with connections to commutative algebra.

**Anthea Monod ***Columbia University, New York*

Time: Tuesday, July 10, 2018, 11:00

**Tropical Statistics & Geometry of Phylogenetic Tree Spaces**

Phylogenetic trees are the fundamental representation of evolutionary processes, and are particularly essential in modeling many important and diverse biological phenomena, such as speciation, the spread of pathogens, and the evolution of cancer. Trees may be compared with one another in a moduli space, whose geometry is determined by a metric. The classical representation of this space is the BHV tree space, endowed with the geodesic metric.

In this talk, I will discuss an alternative distance function, known as the tropical metric. I will present a comparison of geometric and topological properties of tree spaces under the two metrics that are particularly relevant for statistical analysis. I will make the case that the tropical moduli space of phylogenetic trees is a natural setting for probability and statistics, because it allows for a tropical interpretation of linear algebra, which is the basis of classical statistical analysis.

This is joint work with Bo Lin and Ruriko Yoshida.

**Dhruv Ranganathan ***Massachusetts Institute of Technology*

Time: Monday, July 09, 2018, 09:30

**Logarithmic Gromov-Witten theory with expansions**

Gromov-Witten invariants are intersection numbers on spaces of maps to a smooth and compact variety. They are modeled on counts of algebraic curves in a fixed variety. A basic question in Gromov-Witten theory is to understand how these invariants behave in normal crossings degenerations. Logarithmic Gromov-Witten theory is a natural framework for this, and the foundations of this theory have been laid by Abramovich, Chen, Gross, Siebert, Marcus, and Wise in the last decade. Tropical curves and maps organize the geometry of the relevant logarithmic moduli spaces. In this talk, I will discuss ongoing work that completes a proof of the degeneration formula, which expresses the Gromov-Witten invariants of a variety in terms of the invariants of the torus bundles over the strata of the degeneration.

**Felipe Rincon ***University of Oslo*

Time: Monday, July 09, 2018, 15:00

**Tropical Ideals**

I will talk about tropical ideals, which are ideals over the tropical semiring in which any bounded-degree piece is “matroidal”. I will present some of their main properties, together with new results studying the possible varieties they define.

**Louis Rowen ***Bar-Ilan University*

Time: Monday, July 09, 2018, 13:50

**A general algebraic structure theory for tropical mathematics**

The goal of this overview is to present an axiomatic algebraic theory which unifies, simplifies, and ``explains'' aspects of idempotent algebra, tropical algebra, hyperfields, and fuzzy rings, terms of familiar classical algebraic concepts. It was motivated by an attempt to understand whether or not it is coincidental that basic algebraic theorems are mirrored in supertropical algebra (in work with Izhakian and Knebusch), and was spurred by the realization that some of the same results have been obtained in parallel research on hyperfields and fuzzy rings. Our objective is to hone in on the precise axioms that include these various examples, formulate the axiomatic structure, and describe its uses for linear algebra, exterior algebras, geometry, homology. Collaborators in this project so far include Akian, Gatto, Gaubert, Jun, and Mincheva.

**Ben Smith ***Queen Mary University of London*

Time: Tuesday, July 10, 2018, 13:00

**Matching fields and tropical hyperplane arrangements**

An (n,d)-matching field is a collection of matchings such that there is a unique matching for each d-subset of [n]. They naturally arise as minimal matchings of a weighted complete bipartite graph, and can be thought of as a matroid-like structure for tropical geometry. They have relationships to multiple combinatorial objects, in particular arrangements of tropical hyperplanes. In this talk, we show how machinery from tropical geometry can be applied to matching fields via this relationship, and use it to prove two outstanding conjectures of Sturmfels and Zelevinsky on matching fields.

**Ngoc Tran ***University of Texas at Austin*

Time: Tuesday, July 10, 2018, 09:30

**Tropical Gaussians**

Three months ago Bernd Sturmfels asked me "What would be the analogue of the Gaussian distribution in tropical statistics?" I let out a sigh, and before I could give him a lecture, Skype hung up on me. Three months on, and here I am to deliver the answer. That is, in this lecture, I will give and review several recipes for generalizing the Gaussian measure to tropical world, and point out various open research directions.

## Date and Location

**July 09 - 10, 2018**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Yue Ren**

MPI for Mathematics in the Sciences (Germany)**Martin Ulirsch**

Goethe Universität Frankfurt

## Administrative Contact

**Saskia Gutzschebauch**

MPI für Mathematik in den Naturwissenschaften

Contact by Email