3. Postdoctoral Program - MSRI

REPORT ON THE SEMESTER PROGRAM
“TROPICAL GEOMETRY”
EVA-MARIA FEICHTNER, ILIA ITENBERG,
GRIGORY MIKHALKIN AND BERND STURMFELS
MSRI BERKELEY, FALL 2009
Contents
1. Introduction 1
2. Participants 2
3. Organizatorial Structure 2
4. Synergistic Activities 2
5. Workshops 2
6. Research Developments 5
7. Postdocs 7
8. Graduate Students 14
9. Nuggets and Breakthroughs 16
1. Introduction
Recent years have seen a tremendous development in Tropical Geometry that both established
the field as an area of its own right and unveiled its deep connections to numerous branches of
pure and applied mathematics. Formally speaking, Tropical Geometry is the algebraic geometry
over the tropical semiring R ∪ {∞} with operations x⊕y := min{x, y} and x⊙y := x+y. From an
algebraic geometric point of view, algebraic varieties over a field with non-archimedean valuation
are replaced by polyhedral complexes, thereby retaining much of the information about the
original varieties. From the point of view of complex geometry, the geometric combinatorial
structure of tropical varieties is a maximal degeneration of a complex structure on a manifold.
The tropical transition from the objects of algebraic geometry to the polyhedral realm opens
classical problems to a completely new set of techniques, and has already led to remarkable results
in Enumerative Algebraic Geometry, Symplectic Geometry, Dynamical Systems and Computational
Commutative Algebra, among other fields, and to applications in Algebraic Statistics,
Mathematical Biology, and Statistical Physics. Recent tropical papers explore connections to
Low-dimensional Topology, Number Theory, Representation Theory, Optimization, Enumerative
Combinatorics, Stochastic Processes, Random Matrix Theory, and Mathematical Physics.
The semester program at MSRI was the first major research program at a mathematics institute
devoted entirely to Tropical Geometry. Preceded by a number of conferences, workshops
and summer schools at major research centers around the world, the program was the culmination
point of the vivid activities in this newly emerging field, and will be a milestone on its way
to a recognized discipline that straddles Algebra, Analysis, Combinatorics and Geometry.
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