Tropical Geometry and its applications

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.

22 pages, 9 figures, version 2 incorporates corrections in the page proofs

References in corpus (2)

• Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry
• Toric degenerations of toric varieties and tropical curves

Cited by in corpus (84)

• Tropical fans and the moduli spaces of tropical curves
• On the tropical Torelli map
• Moduli spaces of rational tropical curves
• Torelli theorem for graphs and tropical curves
• Formes différentielles réelles et courants sur les espaces de Berkovich
• Rank of divisors on tropical curves
• Unquenched flavor and tropical geometry in strongly coupled Chern-Simons-matter theories
• Lifting tropical intersections
• Hyperfields for Tropical Geometry I. Hyperfields and dequantization
• Brief introduction to tropical geometry
• Algebraic and tropical curves: comparing their moduli spaces
• The diagonal of tropical matroid varieties and cycle intersections
• Intersecting Solitons, Amoeba and Tropical Geometry
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• A Field of Generalised Puiseux Series for Tropical Geometry
• Geometry of tropical moduli spaces and linkage of graphs
• Stable Intersections of Tropical Varieties
• A tropical approach to non-archimedean Arakelov theory
• Correspondence theorems for tropical curves I
• Bitangents of tropical plane quartic curves
• Lifting representations of finite reductive groups I: Semisimple conjugacy classes
• Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
• A-Tint: A polymake extension for algorithmic tropical intersection theory
• Completions, Reversals, and Duality for Tropical Varieties
• Lifting tropical bitangents
• Realizability of tropical canonical divisors
• On rational equivalence in tropical geometry
• Phase tropical hypersurfaces
• Generic Tropical Varieties
• Universal algorithms for solving the matrix Bellman equations over semirings
• Euler Characteristic of real nondegenerate tropical complete intersections
• Tropical Intersection Theory from Toric Varieties
• Balancing conditions in global tropical geometry
• The j-invariant of a plane tropical cubic
• Equations for Chow and Hilbert Quotients
• Idempotent and tropical mathematics and problems of mathematical physics (Volume II)
• On Block-Goettsche multiplicities for planar tropical curves
• Tropical geometry of genus two curves
• Combinatorics and real lifts of bitangents to tropical quartic curves
• Construction of schemes over $F_1$, and over idempotent semirings: towards tropical geometry
• Log-concavity of characteristic polynomials and the Bergman fan of matroids
• Anticanonical tropical cubic del Pezzos contain exactly 27 lines
• A Tropical Computation of Refined Toric Invariants
• Specialization of linear systems from curves to graphs
• Tropical geometry and correspondence theorems via toric stacks
• Tropical curves and covers and their moduli spaces
• Quasi--Euclidean classification of Alcoved Convex Polyhedra
• SYZ mirror symmetry for toric varieties
• Commutative $\nu$-algebra and supertropical algebraic geometry
• A tropical intersection product in matroidal fans
• The Newton polygon of a planar singular curve and its subdivision
• Classification of normal toric varieties over a valuation ring of rank one
• On refined count of rational tropical curves
• Topology of moduli spaces of tropical curves with marked points
• Real phase structures on matroid fans and matroid orientations
• Dimension counts for singular rational curves via semigroups
• Tropical compound matrix identities
• A guide to tropical modifications
• Toric and tropical compactifications of hyperplane complements
• Some statistics about Tropical Sandpile Model
• The tropical $j$-invariant
• Exact Sequences in Non-Exact Categories (An Application to Semimodules)
• A combinatorial Li-Yau inequality and rational points on curves
• Hyperelliptic graphs and the period mapping on outer space
• Tropical Nevanlinna theory and ultra-discrete equations
• Welschinger invariants of small non-toric Del Pezzo surfaces
• Superforms, supercurrents, minimal manifolds and Riemannian geometry
• Dupin cyclides osculating surfaces
• Moduli spaces of tropical curves of higher genus with marked points and homotopy colimits
• How to repair tropicalizations of plane curves using modifications
• Combinatorics and Genus of Tropical Intersections and Ehrhart Theory
• Tropical flag varieties
• On Tropical Linear and Integer Programs
• Idempotent and tropical mathematics. Complexity of algorithms and interval analysis
• Resultants over Commutative Idempotent Semirings
• Automorphism Groups on Tropical Curves: Some Cohomology Calculations
• Degenerations of amoebae and Berkovich spaces
• Tropical orbit spaces and the moduli spaces of elliptic tropical curves
• Rank of tropical curves and tropical hypersurfaces
• Number of common roots and resultant of two tropical univariate polynomials
• Tropical Theta Functions and Riemann-Roch Inequality for Tropical Abelian Surfaces
• Tropical mathematics, classical mechanics and geometry
• Minkowski weights and the Grothendieck group of a toric variety
• Constructing polynomial systems with many positive solutions using tropical geometry