#### Workshop Tropical Doener, FU Berlin, May 27-28, 2015

The two-day workshop called tropical doener on tropical geometry and related subjects is supposed to bring the German and local-European researchers in the area together to exchange latest ideas.

Please also note the kickoff workshop on NonLinear Algebra taking place in Berlin May 29-30. Furthermore, on May 28 there will be the Einstein public lecture called Museen, Dreiecke und algebraische Kurven by Michael Joswig.

<!-- We gratefully acknowledge support by the DFG priority program SPP 1489 on [[http://www.computeralgebra.de|Computeralgebra]]. -->

##### Preliminary Schedule

The talks take place in Dahlem.

##### Wednesday, May 27

Arnimallee 6, Room 031

 15:30 16:30 Bernd Sturmfels Tropical Plane Curves coffee Ngoc Tran Tropical geometry and Auction Theory Food and music TBA
##### Thursday, May 28

ZIB Seminar Room

 9:00 10:00 Frederic Bihan Sharp fewnomial bounds for tropical polynomial systems coffee Anna Lena Birkmeyer Realizability of Tropical Curves in a Plane Simon Hampe Tropical computations in polymake lunch Hannah Markwig Tropicalizing rational relative Gromov-Witten theory of P¹
##### Abstracts

Bernd Sturmfels Tropical Plane Curves

Abstract: Tropical geometry is a combinatorial shadow of classical geometry. Algebraic curves in the tropical plane are dual to triangulations of convex polygons. We discuss the intrinsic geometry of these objects, with focus on the moduli space of metric graphs that represents tropical plane curves.

Ngoc Tran Tropical geometry and Auction Theory

Abstract: In 2012, two economists Baldwin and Klemperer at Oxford give a sufficient condition for the existence of equilibrium in product-mix auctions. This unearths an interesting connection between tropical geometry and economics, specifically, auction theory, and more generally, general equilibria for indivisible goods. In this talk I will introduce auction theory for a mathematical audience and give a proof of the Baldwin–Klemperer result via tropical geometry. We also outline a number of interesting open problems. No background knowledge in economics is required.

Frederic Bihan Sharp fewnomial bounds for tropical polynomial systems

Abstract: We introduce the discrete mixed volume of finite sets W_1,…,W_n of the affine $n$-space. We show that this quantity is a sharp bound on the number of non degenerate solutions of a tropical polynomial system with given supports. The proof is based on a irrational mixed decomposition technique applied to the Newton polytopes. As a consequence, we obtain that a conjectural bound proposed long time ago by Kouchnirenko for real polynomial systems works for tropical polynomial systems.

Anna Lena Birkmeyer Realizability of Tropical Curves in a Plane

Abstract: Given a 2-dimensional matroid fan F and an algebraic plane E tropicalizing to F, we investigate the question which tropical curves in F, i.e., balanced weighted 1-dimensional polyhedral complexes, arise as the tropicalization of algebraic curves in E. We use projections to the plane to present an algorithm able to decide for any given tropical curve if it is realizable, i.e. if it is the tropicalization of an algebraic curve in E. Moreover, we give sufficient and necessary conditions to realizability and describe the space of all realizable curves.

Simon Hampe Tropical computations in polymake

Abstract: In this presentation I'll showcase some of the features of polymake that deal with tropical things. Most of these are relatively new or have been recently revamped, such as:

• Tropical arithmetic (Min or Max - it's your choice!)
• Tropical convex hull computations
• Tropical polynomials and tropical hypersurfaces
• My own extension „a-tint“, which mostly deals with tropical

intersection theory (but has a much wider range of features than that. For an overview of the old version see https://bitbucket.org/hampe/atint).

Hannah Markwig Tropicalizing rational relative Gromov-Witten theory of P^1

Abstract: We show that the relative stable map compactification of M_0,n (for maps to P^1, relative to two points) is a tropical compactification. Furthermore, the tropicalization of the open part equals the tropical space of relative stable maps to P^1. Consequently, the Chow ring of the relative stable map space can be computed by means of tropical intersection theory in an intuitive way. This is joint work with Renzo Cavalieri and Dhruv Ranganathan.