  
E.M. Feichtner

A combinatorial introduction to tropical geometry


Abstract:
The core undertaking of tropical geometry is to transform
algebrogeometric objects into piecewise linear ones, while
retaining much of the original algebraic information. This, in
particular, opens problems on algebraic varieties to a completely new
set of techniques from the discrete geometric realm. Tropical
geometry draws from the results of geometric combinatorics and, at the
same time, exhibits a wealth of interesting structures to be explored
from the discrete geometric viewpoint.
In this series of lectures we provide an introduction to tropical
geometry from the combinatorial point of view. We will keep the
exposition selfcontained by introducing the necessary concepts
from geometric combinatorics as they cross our way.



I. Itenberg

Enumerative tropical geometry


Abstract:
The purpose of the minicourse is to make an introduction to tropical
geometry with an emphasis on the applications of tropical geometry
to complex and real enumerative problems. Tropical geometry has
deep and important relations with many branches of mathematics. An
important link between the complex algebraic world and the
tropical one is given by Mikhalkin's correspondence theorem. We
discuss various versions of correspondence theorems, which
together with Welschinger's discovery of a real analog of genus
zero GromovWitten invariants produce new results
on enumeration of real rational curves.



V. Kharlamov

Selected topics in topology of real algebraic varieties


Abstract:
In this minicourse we will treat certain
topological properties and characteristics of real algebraic varieties
making emphasis on the Betti numbers. In particular,
we will give some explicite upper bounds for the Betti numbers
of projective varieties and will discuss the optimality
of these estimates.



F. Sottile

Bounds for real solutions to polynomial systems


Abstract:
Understanding, finding, or even deciding the existence of real solutions to
a system of equations is a very difficult problem with many applications. While
it is hopeless to expect much in general, we know a surprising amount about these
questions for systems which possess additional structure.
We will focus on equations from toric varieties and homogeneous spaces, particularly
Grassmannians. Not only is much known in these cases, but they encompass some of the
most common applications. The results we discuss may be grouped into two themes:
(1) Upper bounds on the number of real solutions
(2) Lower bounds on the number of real solutions
Upper bounds as in (1) bound the complexity of the set of real solutions.
Lower bounds as in (2) give an existence proof for real solutions.
