|E.-M. Feichtner||A combinatorial introduction to tropical geometry|
|16:00--17:30 Wednesday, Thursday, and Friday|
|I. Itenberg||Enumerative tropical geometry|
|16:00--17:30 Monday, 14:00--15:30 Tuesday, 10:30--12:00 Thursday and Friday|
|V. Kharlamov||Selected topics in topology of real algebraic varieties|
|10:30--12:00 Monday, Tuesday, and Wednesday.|
|F. Sottile||Bounds for real solutions to polynomial systems|
|16:00--17:30 Tuesday and 14:00--15:30 Wednesday, Thursday, and Friday.|
|E.-M. Feichtner||A combinatorial introduction to tropical geometry||
The core undertaking of tropical geometry is to transform
algebro-geometric 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 self-contained by introducing the necessary concepts from geometric combinatorics as they cross our way.
|I. Itenberg||Enumerative tropical geometry||Abstract: The purpose of the mini-course 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 Gromov-Witten invariants produce new results on enumeration of real rational curves.|
|V. Kharlamov||Selected topics in topology of real algebraic varieties||Abstract: In this mini-course 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||
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.