This two day event features new developments in tropical geometry and its applications, and it offers a perspective on future directions. Time is also provided for participants to discuss topics of common interest and current problems, with the aim of fostering new research connections.

Travel funding and accommodation can be provided for early-career participants such as postdoctoral researchers and PhD students. Applicants are expected to hand in a short letter of motivation, CV as well as a recommendation letter and to kindly agree to present their work in the form of a poster. After registering you will receive further instructions on the application process by email. The deadline for funding applications is May 21, 2018.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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 $F_d( trop X)$ and I will show its relation with the tropicalization $trop F_d(X)$ of the classical Fano scheme. In particular I will focus on the tropical Fano schemes of tropicalized linear spaces and tropicalized toric varieties.

Participants

Hulya Arguz

Imperial College London

Amanda Cameron

MPI MIS

Türkü Özlüm Çelik

Max Planck Institute for Mathematics in the Sciences

Saurav Dwivedi

Masaryk University, Czech Republic

Marzieh Eidi

Max Planck Institute

Gavril Farkas

Humboldt-Universität zu Berlin

Tyler Foster

Max-Planck-Institut für Mathematik Bonn

Francesco Galuppi

MPI MiS

Jeffrey Giansiracusa

Swansea University

Alexander Guterman

Lomonosov Moscow State University

Michael Joswig

TU Berlin

Michael Kemeny

Stanford University

Khazhgali Kozhasov

MPI MIS

Sara Lamboglia

Goethe University Frankfurt

Gaku Liu

MPI MiS

Georg Loho

Ecole polytechnique fédérale de Lausanne

Robert Löwe

TU Berlin

Artem Maksaev

Lomonosov Moscow State University

Anthea Monod

Columbia University, New York

Raffaella Mulas

Max Planck Institute for Mathematics in the Sciences

Nina Otter

Max Planck Institute for Mathematics in the Sciences

Marta Panizzut

TU Berlin

Dhruv Ranganathan

Massachusetts Institute of Technology

Yue Ren

Max Planck Institute for Mathematics in the Sciences

Felipe Rincon

University of Oslo

Louis Rowen

Bar-Ilan University

Mahsa Sayyary Namin

MPI MIS

Emre Sertöz

MPI MiS

Pavel Shteyner

Lomonosov Moscow State University

Thomas Skill

University of Applied Sciences Bochum

Ben Smith

Queen Mary University of London

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Sascha Timme

TU Berlin

Ngoc Tran

University of Texas at Austin

Martin Ulirsch

Goethe Universität Frankfurt

Alejandro Vargas

Universität Bern

Haopeng Wang

KU Leuven

Charles Wang

UC Berkeley

Haopeng Wang

KU Leuven

Dmitry Zakharov

Central Michigan University

Leon Zhang

University of California, Berkeley

Scientific Organizers

Yue Ren

Max Planck Institute for Mathematics in the Sciences, Germany

Martin Ulirsch

Goethe Universität Frankfurt

Administrative Contact

Saskia Gutzschebauch

Max-Planck-Institut für Mathematik in den Naturwissenschaften
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