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.
The seminar takes place in the mathematics buildings E 2 4 resp. E 2 5.
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|
|17:00||Ilia Zharkov||When tropical Prym is a Jacobian|
|18:00||Pizza, drinks, music UND FUSSBALL|
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|
|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|
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.
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.
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.
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.
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.