Oleg Viro, Steklov Insitute of Mathematics (St Petersburg) and SUNY Stony Brook
Patchworking of algebraic varieties and tropical geometry

Algebraic Geometry has a piecewise linear core visible in logarithmic coordinates. This core was recently named Tropical Geometry. The relation between Tropical and Algebraic Geometries is similar to that between Classical and Quantum Mechanics. Deformations similar to quantization turn tropical varieties into usual complex or real algebraic varieties. This is used as a powerful way to construct algebraic varieties with interesting properties. As long as we are interested in topological or other coarse properties of real algebraic varieties, we can build varieties by patchworking, a sort of "cut and paste" technique. If the building blocks are essentially linear, the patchworking can be considered as a deformation of a tropical variety to a real one.

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