3. Postdoctoral Program - MSRI

REPORT ON THE MSRI INTRODUCTORY WORKSHOP
OF THE PROGRAM ”TROPICAL GEOMETRY”
August 24 - 28, 2009
Organizers: Eva-Maria Feichtner, Ilia Itenberg, Grigory Mikhalkin and Bernd Sturmfels
1. Scientific description
The main purpose of this workshop was to lay the foundations for the beginning MSRI fall program
“Tropical Geometry.” Tropical Geometry is a branch of geometry that has appeared just
recently. Formally, it can be viewed as Algebraic Geometry over the semiring of tropical numbers.
The tropical numbers are the real numbers enhanced with negative infinity and equipped
with two arithmetic operations called tropical addition and tropical multiplication. The tropical
addition is the operation of taking the maximum. The tropical multiplication is the conventional
addition. These operations are commutative, associative and satisfy the distribution law. The
term “tropical” was coined by computer scientists and is a tribute to Brazil, in particular the
contributions of the Brazilian mathematician Imre Simon to the theory of formal languages.
It turns out that tropical algebra describes some meaningful geometric objects, namely, Tropical
Varieties. From the topological point of view tropical varieties are polyhedral complexes
equipped with a particular geometric structure coming from tropical algebra. From the point
of view of Complex Geometry this geometric structure is the worst possible degeneration of
complex structure on a manifold. From the point of view of Symplectic Geometry a tropical
variety is the result of a Lagrangian collapse of a symplectic manifold along a singular fibration
by Lagrangian tori.
Tropical Geometry has applications in Real Algebraic Geometry, Enumerative Geometry,
Mirror Symmetry, Symplectic Geometry, as well as Combinatorial and Computational Geometry,
while the list of its applications keeps growing. E.g., Tropical Geometry made a most recent
and brand-new appearance in the Statistical Physics work of R. Kenyon and A. Okounkov where
they studied mathematical models for dimers accumulation. Currently there are several research
groups around the globe who are doing active research in Tropical Geometry from somewhat
different points of view.
With this introductory workshop we were able to portray a substantial part of this evolving
field through mini-courses and through complementing research talks, thereby providing both
an entrance point for newcomers to the field and a point of outset for those who came to attend
the program.
In five mini-courses of three 1-hour lectures each, the foundational aspects of Tropical Geometry
were covered as well as its connections with adjacent areas: Algebraic Geometry, Geometric
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