Mini-cours (3 exposés d'1 h 30)

Ilia Itenberg (Strasbourg)
Interactions between tropical geometry and real enumerative geometry

The main purpose of these lectures is to present several applications (based on Mikhalkin's correspondence theorem) of tropical geometry in real enumerative geometry. In particular, we will discuss different combinatorial and recursive formulas designed for a calculation of Welschinger invariants, the invariants which can be viewed as real analogs of Gromov-Witten invariants.

Grigory Mikhalkin (Toronto)
Basic notions and applications of tropical geometry

These three lectures will start from some foundational tropico-geometric definitions, such as that of tropical varieties and their equivalences. The emphasis will be made on tropical curves and their moduli spaces. In the final part of the course we will review applications to some enumerative problems of classical algebraic geometry.

Mikael Passare (Stockholm)
Basic amoeba theory

This series of four lectures is intended to give a quick, but also quite broad, introduction to the properties and applications of mathematical amoebas. The focus will primarily be on complex hypersurface amoebas. Below is a list containing some of the topics that we plan to cover. Amoebas and coamoebas, definitions and examples, relations to Laurent series and Mellin transforms. Ronkin functions, amoeba spines, relations to tropical geometry. Amoeba contours, logarithmic Gauss maps, relations to discriminants and hypergeometric functions. Further applications of amoebas.

Eugenii Shustin (Tel Aviv)
Tropical curves and algebraic curves: correspondence theorems

We present various correspondence theorems between plane tropical curves and algebraic curves on surfaces, both real and complex, which constitute a core for applications of the tropical geometry to the complex and real enumerative algebraic geometry, notably, computation of Gromov-Witten and Welschinger invariants of surfaces.

Oleg Viro (Uppsala)

Gluing of spaces, a common and easy operation in Topology, is not that easy it algebraic geometry due to higher rigidity of algebraic variety. It becomes easier if appears in continuous families. Patchworking is a special kind of gluing that can be applied to several algebraic varieties and includes their union into a family of varieties in which a generic member topologically is the result of gluing of the initial varieties. Various versions of the patchworking construction will be presented. We will start with perturbations of semi-quasi-homogeneous singularities, an proceed through general patchworking of hypersurfaces to combinatorial patchworking and tropical hypersurfaces. Then patchworking of other objects will be discussed.

Tropical computer science

The goal of this mini-course is to give a taste of the use and pertinence of the tropical semiring in theoretical computer science. We introduce the notion of a max-plus automaton, or rational series with non-commuting variables and multiplicities in the tropical semiring. We illustrate the various facets of the object, which is a tool for tackling decidability issues on formal languages as well as a tool for evaluating the performances of systems. We discuss the subclass of Tetris automata for which a simple geometric representation exists. After this preamble, we focus on the issue of system modelling, and we motivate the variations and extensions of tropicality appearing in this context. Some details are given in the two abstracts below.

Jean Mairesse (Cnrs - Paris 7)
Max-plus modelling and analysis of synchronised networks
Max-plus linear functions, and more generally max-plus automata, appear naturally in the modelling of the timed behavior of networks with synchronizations and delays. We consider in particular the model of timed event graphs, and the one of Tetris heaps. To solve optimisation problems for such networks, one is led to introduce a generalisation: topical maps and iterated systems of such maps.

Thierry Bousch (Cnrs - Orsay)
On retarders
I propose a new model for delays, inspired by topical functions and relevant for stochastic discrete event systems. Retarders are stochastic increasing functions from Z to Z or R to R whose distribution is invariant under conjugation by translations. They can model "First In First Out" delay mechanisms like the ones occuring in the subway or in traffic lights, and they are natural candidates to represent stochastic delays in event graphs.

Mikhail Kapranov (Yale)
Non-archimedean Amoebas