## Tropical Dibbelabbes in Saarbrücken

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

We gratefully acknowledge support by the DFG priority program SPP 1489 on Computeralgebra.

##### Speakers

- Timo de Wolff
- Andreas Gross
- Milena Hering
- Michael Joswig
- Ilia Zharkov

##### Preliminary Schedule

The seminar takes place in the mathematics buildings E 2 4 resp. E 2 5.

##### Thursday, June 26

The talks take place in Hoersaal IV in E 2 4. This is the lecture room on the first floor of E 2 4. Turn right after you enter the building.

15:30 | Milena Hering | Projective normality of tropical curves |
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16:30 | coffee | |

17:00 | Ilia Zharkov | When tropical Prym is a Jacobian |

18:00 | Pizza, drinks, music UND FUSSBALL |

##### Friday, June 27

The talks take place in U 39 (Zeichensaal) in E 2 5. This is a classroom on the lower level of the building E 2 5. It faces towards the green area between E 2 4 and E 2 5.

9:00 | Michael Joswig | Tropical Linear Programming |
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10:00 | coffee | |

10:30 | Andreas Gross | Correspondence Theorems via Tropicalizations of Moduli Spaces |

11:35 | Timo de Wolff | Amoebas, Nonnegative Polynomials and Sums of Squares Supported on Circuits |

##### Abstracts

##### Projective normality of tropical curves

It is a classical theorem that embeddings of algebraic curves induced by line bundles of sufficient high degree are projectively normal, i.e., the intersections of the hypersurfaces of a fixed degree with the curve form a complete linear system. In this talk I will discuss possible notions of projective normality for embeddings of tropical curves as well as some first results. This is joint work with Josephine Yu.

##### When tropical Prym is a Jacobian

I will follow the 3 classical cases of Mumford: hyperelliptic, trigonal and plane quintic curves in the tropical setting. Plus one of Beauville: the Wirtinger cover.

##### Tropical Linear Programming

Looking at fields of formal Puiseux series with real coefficients provides a fairly natural approach to carry over linear programming cencepts and methods to tropical polyhedra. It is less obvious that this is fruitful for a number of reasons. The purpose of this talk is to show how ideas from tropical geometry can be exploited to address several issues concerning the algorithmic complexity of classical linear programming. In this way, in particular, we relate the classical simplex method to a well-studied decision problem whose complexity is in NP \cap co-NP. Moreover, we construct classical linear programs with a central path of large total curvature; this disproves a conjecture of Deza, Terlaky and Zinchenko. Joint work with Xavier Allamigeon, Pascal Benchimol and Stephane Gaubert.

##### Correspondence Theorems via Tropicalizations of Moduli Spaces

Since its emergence about a decade ago, many new techniques have been developed in tropical enumerative geometry. In this talk I will discuss how our current knowledge of tropical moduli spaces, their intersection theory, and the tropicalization map can be used to obtain correspondence theorems for rational curves in toric varieties.

##### Amoebas, Nonnegative Polynomials and Sums of Squares Supported on Circuits

We completely charaterize sections of the cones of nonnegative polynomials and sums of squares with polynomials supported on circuits – a genuine class of sparse polynomials. In particular, nonnegativity is characterized by an invariant, which can be immediately derived from the initial polynomial via using a new norm based relaxation strategy. Based on these results, we obtain a completely new class of nonnegativity certificates independent from sums of squares certificates. Furthermore, nonnegativity of such polynomials f coincides with solidness of the amoeba of f, i.e., the Log-absolute-value image of the algebraic variety V(f) of f in the torus. These results establish a first direct connection between amoeba theory and nonnegativity of polynomials. They generalize earlier works both in amoeba theory and real algebraic geometry by Fidalgo, Kovacec, Reznick, Theobald and myself. This talk is based on joint work with Sadik Iliman.

##### Registration

- Please send an email to Christine Wilk (charlyATmath.uni-sb.de) until May 20.

##### Accomodation

- Hotel Stadt Hamburg is close to the train station.
- Hotel Madeleine is in the city, about 15 minutes walking distance from the train station.
- Motel One is in the city, about 20 minutes walking distance from the train station.
- If you mention that you go to the University, they might offer a special price.

##### How to get here

- The train schedule can be found here: trains.
- You can take bus 124 from the train station (Hauptbahnhof) to the university restaurant (Universitaet Mensa) or to the university bus terminal (Universitaet Busterminal - this is the final stop). The bus schedule can be found here.
- If you'd like to go from the city to the university, you can take the buses 101, 102, 103, 109 and 111 from Rathaus (city hall) to Universitaet Mensa or Universitaet Busterminal: use the Fahrplanauskunft (in pink on the page) to find the departure time of the next bus.
- The bus fee can be paid when entering the bus (you need Euro cash, but not the exact fare). It's 2,50 € for a single ride and 2,10 € with a BahnCard.
- A map of campus can be found here. The bus stop Universitaet Busterminal is the little green-yellow-white H in front of the big parking in front of E3 1.
- The mathematics building is E2 4. Our offices are on second floor (301-305).

##### Organizers

- Christian Haase
- Hannah Markwig