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.

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.

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.

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.

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.

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.