By Ricardo Castano-Bernard, Fabrizio Catanese, Maxim Kontsevich, Tony Pantev, Yan Soibelman, Ilia Zharkov
The dating among Tropical Geometry and replicate Symmetry is going again to the paintings of Kontsevich and Y. Soibelman (2000), who utilized equipment of non-archimedean geometry (in specific, tropical curves) to Homological reflect Symmetry. together with the following paintings of Mikhalkin at the “tropical” method of Gromov-Witten idea and the paintings of Gross and Siebert, Tropical Geometry has now turn into a strong device. Homological reflect Symmetry is the realm of arithmetic centred round a number of specific equivalences connecting symplectic and holomorphic (or algebraic) geometry. The primary rules first seemed within the paintings of Maxim Kontsevich (1993). approximately conversing, the topic could be approached in methods: both one makes use of Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow photograph, extra constructed by way of Kontsevich and Soibelman) or one makes use of Lefschetz fibrations of symplectic manifolds (suggested via Kontsevich and extra constructed via Seidel). Tropical Geometry reports piecewise-linear items which look as “degenerations” of the corresponding algebro-geometric objects.