# Berlin, March 30-31st, 2015

## Math Institute - Technische Universität Berlin

# Program

All talks will take place in MA 041 of the math building of the TU Berlin

## Monday, March 30th

** 11.00 Coffee
**

11.30 ** Michael Joswig ** (TU Berlin)

Weighted digraphs and tropical cones.

#### This talk is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, exterior descriptions in terms of tropical halfspaces enter the picture. Our method is to employ ordinary convex polyhedra naturally associated with weighted digraphs. This way we can relate to and use results from classical combinatorial optimization. We arrive at generalizations of several results of Develin & Sturmfels (2004). Further, we solve a conjecture of Develin & Yu (2007). Joint work with Georg Loho.

14.00 ** Nikita Kalinin ** (Université de Genève)

Tropical legendrian curves.

We consider algebraic curves in CP^3, which are tangent to the form ydx-xdy+wdz-zdw. I survey what is known (not a lot) about such curves, and then we classify them up to degree 3. The tropical analogs of legendrian curves are more interesting and enjoys some combinatorial properties.

** 15.00 Coffee
**

** 15.30 Ferit Öztürk (Bogaziçi University)
**

** Real tight lens spaces and links of real surface singularities
**

We will present the classification of the real tight contact structures on lens spaces up to equivariant contact isotopy. Our main tools are real convex surface theory, tomography and the real tight links of real surface singularities. Through the classification we will observe that on each lens space there is a unique real tight structure which does not admit a genus-1 real Heegaard decomposition and that the ones which come from a genus-1 real Heegaard decomposition are very few compared to the (not necessarily real) tight structures.

## Tuesday, March 31st

** 10.00 Lionel Lang (Université de Genève)
**

**
Simples Harnack curves and their relatives.
**

Simple Harnack curves have been introduced by Mikhalkin as smooth real algebraic curves sitting in maximal position in toric surfaces. After recalling their most interesting features, we will give an easy generalization to the case of singular curves. We will show how tropical geometry helps constructing many new examples and give their first properties. We will insist on their very tropical nature, first discovered by Kenyon and Okounkov. If time permits, we will relate simple Harnack curves to questions on coamoebas, introducing multi-Harnack curves.

** 11.00 Coffee
**

** 11.30 Ilia Zharkov (Kansas State University)
**

**
Double covers of tropical plane quintics and their Pryms.
**

A classical result of Mumford says that the Prym of a double covering of a smooth plane quintic is a Jacobian if and only if the cover corresponds to an even theta characteristic. I will give the tropical version of this result.

** 14.00 Gavril Farkas (HU Berlin)
**

**The uniformization of the moduli space of abelian varieties of
dimension six.
**

The general principally polarized abelian variety of dimension at most five is known to be a Prym variety. This reduces the study of abelian varieties of small dimension to the beautifully concrete theory of algebraic curves. I will discuss recent progress on finding a structure theorem for principally polarized abelian varieties of dimension six, and the implications this uniformization result has on the geometry of the moduli space. The main result is that a general ppav of dimension 6 is a Prym-Tyurin variety cover corresponding to an E_6-cover of a projective line. The proof uses a maximal degeneration which has a lot in common with tropical geometery.