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Tropical geometry and its applications
 the Proceedings of the Madrid ICM
"... Abstract. These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM. 1. ..."
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Cited by 142 (7 self)
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Abstract. These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM. 1.
From real affine geometry to complex geometry
"... Abstract. We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of CalabiYau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical orderbyorder descrip ..."
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Cited by 60 (8 self)
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Abstract. We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of CalabiYau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical orderbyorder description of the degeneration via families of tropical trees. This gives complete control of the Bmodel side of mirror symmetry in terms of tropical geometry. For example, we expect our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods. This
Tropical fans and the moduli spaces of tropical curves
 COMPOS. MATH
, 2009
"... We give a rigorous definition of tropical fans (the “local building blocks for tropical varieties”) and their morphisms. For such a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does ..."
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Cited by 38 (2 self)
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We give a rigorous definition of tropical fans (the “local building blocks for tropical varieties”) and their morphisms. For such a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point — a statement that can be viewed as the beginning of a tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some Rr) together with the evaluation and forgetful morphisms. Using our results this gives new, easy, and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any Rr) through given points are independent of the points.
Logarithmic GromovWitten invariants
 Journal of the American Mathematical Society
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The tropical vertex
"... Abstract. Elements of the tropical vertex group are formal families of symplectomorphisms of the 2dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative GromovWitten invariants of toric surfaces. ..."
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Cited by 34 (11 self)
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Abstract. Elements of the tropical vertex group are formal families of symplectomorphisms of the 2dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative GromovWitten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold GromovWitten theory. Contents
The CaporasoHarris formula and plane relative GromovWitten invariants
 46 Getzler, E.: Intersection theory on M1,4 and elliptic GromovWitten
, 1997
"... Abstract. Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d, g) of complex plane curves of degree d and genus g through 3d + g − 1 general points with the help of relative GromovWitten invariants. Recently, Mikhalkin has found a way to reinterpret the numbers N(d, ..."
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Cited by 33 (9 self)
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Abstract. Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d, g) of complex plane curves of degree d and genus g through 3d + g − 1 general points with the help of relative GromovWitten invariants. Recently, Mikhalkin has found a way to reinterpret the numbers N(d, g) in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes. We relate these two results by defining an analogue of the relative GromovWitten invariants and rederiving the CaporasoHarris formula in terms of both tropical geometry and lattice paths. 1.
The numbers of tropical plane curves through points in general position
 JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK (TO APPEAR)
, 2006
"... We show that the number of tropical curves of given genus and degree through some given general points in the plane does not depend on the position of the points. In the case when the degree of the curves contains only primitive integral vectors this statement has been known for a while now, but t ..."
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Cited by 26 (8 self)
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We show that the number of tropical curves of given genus and degree through some given general points in the plane does not depend on the position of the points. In the case when the degree of the curves contains only primitive integral vectors this statement has been known for a while now, but the only known proof was indirect with the help of Mikhalkin’s Correspondence Theorem that translates this question into the wellknown fact that the numbers of complex curves in a toric surface through some given points do not depend on the position of the points. This paper presents a direct proof entirely within tropical geometry that is in addition applicable to arbitrary degree of the curves.
The StromingerYauZaslow conjecture: From torus fibrations to degenerations
 IN PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS
, 2008
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A tropical toolkit
"... Abstract. We give an introduction to Tropical Geometry and prove some results in Tropical Intersection Theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of degenerations of varieties using projective notnec ..."
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Abstract. We give an introduction to Tropical Geometry and prove some results in Tropical Intersection Theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of degenerations of varieties using projective notnecessarilynormal toric varieties. The second part is a foundational account of tropical intersection theory with proofs of some new theorems relating it to classical intersection theory. 1.