North American Workshop on Tropical Geometry
Ilia Itenberg, Louis Pasteur, Y. Soibelman · 2007
Tropical Geometry is a branch of Geometry that has appeared just recently. Formally, it can be viewed as a sort of Algebraic Geometry with the underlying algebra based on the so-called tropical numbers. The tropical numbers (the term ”tropical” comes from Computer Science and commemorates Brazil, in particular a contribution of the Brazilian school to the language recognition problem) 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. It turns out that such tropical algebra describes some meaningful geometric objects, namely, the Tropical Varieties. From the topological point of view the tropical varieties are piecewise-linear 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 the tropical variety is the result of the Lagrangian collapse of a symplectic manifold (along a singular fibration by Lagrangian tori). The easiest to describe are tropical varieties in dimension 1, i.e. tropical curves. These are the so-called ”metric graphs”, i.e. finite graphs equipped with an inner metric such that all ”leaves”, i.e. the edges adjacent to 1-valent vertices have infinite length. There is a finite-dimensional moduli space of tropical curves once we fix the number of cycles in the graph (this is the tropical counterpart of the genus) and the number of leaves (this is the tropical counterpart of the number of punctures). Such a moduli space is itself a tropical orbifold and there is a certain intersection theory on it. From the point of view of Toric Geometry tropical varieties are limiting shapes of the amoebas of algebraic varieties under the deformation degenerating the argument torus. Such degeneration can be described by varying the base of the logarithm in the amoeba map to infinity. In toric geometry such construction is known as ”the patchworking”, it was introduced by O. Viro in 1979 for the needs of real algebraic geometry to give a way to construct real forms of complex algebraic varieties with controlled topology. Historically this was perhaps the first time of implicit appearance of tropical geometry. Since this appearance there were several discoveries and proposals that stirred the research related to the area, most notably the introduction of amoebas by I. M. Gelfand, M. Kapranov and A. Zelevinski (and, in particular, the introduction of nonArchimedean amoebas by Kapranov), the proposal to use tropical curves in the context of Mirror Symmetry (particularly, for computation of the Gromov-Witten invariants) by M. Kontsevich, the introduction of the Morse category by Fukaya and the introduction of tropical formalism to Computational Algebraic Geometry