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Tropical geometry and its applications. (English) Zbl 1103.14034
Sanz-Solé, Marta (ed.) et al., Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-022-7/hbk). 827-852 (2006).
This survey presents the state of art in the tropical algebraic geometry which can be regarded as algebraic geometry over the tropical semifield, real numbers equipped with operations of maximum and addition. The main objects of tropical algebraic geometry have piecewise-linear nature: tropical varieties are certain finite polyhedral complexes, regular and rational functions are piecewise-linear, tropical morphisms are locally integral-affine maps. The modern development of tropical algebraic geometry includes intersection theory of tropical cycles, moduli spaces of tropical curves, Jacobians and Riemann-Roch theorem for tropical curves. The key relation between the tropical and classical algebraic geometry, representing tropical objects as degenerations of respective classical objects, allows one to successively translate some classical algebraic geometry problems, notably, enumeration of complex and real algebraic curves, into tropical geometry problems which often are easier to solve. The author demonstrates this by presenting tropical solutions to the problems of computing Gromov-Witten and Welschinger invariants of toric varieties, and to the problem of realizing knots in the three-space by real algebraic curves. For the entire collection see [Zbl 1095.00005].

14P99Real algebraic and real analytic geometry
14H50Plane and space curves
14N10Enumerative problems (algebraic geometry)
52B20Lattice polytopes (convex geometry)
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