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 (85)

• 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
• A tropical approach to non-archimedean Arakelov theory
• Geometry of tropical moduli spaces and linkage of graphs
• Stable Intersections of Tropical Varieties
• Correspondence theorems for tropical curves I
• Bitangents of tropical plane quartic curves
• Tropical Arithmetic and Tropical Matrix Algebra
• 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
• Universal algorithms for solving the matrix Bellman equations over semirings
• Generic Tropical Varieties
• 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
• Idempotent and tropical mathematics and problems of mathematical physics (Volume II)
• Equations for Chow and Hilbert Quotients
• Specialization of linear systems from curves to graphs
• A Tropical Computation of Refined Toric Invariants
• On Block-Goettsche multiplicities for planar tropical curves
• Tropical geometry of genus two curves
• Construction of schemes over $F_1$, and over idempotent semirings: towards tropical geometry
• Anticanonical tropical cubic del Pezzos contain exactly 27 lines
• Combinatorics and real lifts of bitangents to tropical quartic curves
• Log-concavity of characteristic polynomials and the Bergman fan of matroids
• Commutative $\nu$-algebra and supertropical algebraic geometry
• Quasi--Euclidean classification of Alcoved Convex Polyhedra
• Tropical curves and covers and their moduli spaces
• SYZ mirror symmetry for toric varieties
• A tropical intersection product in matroidal fans
• Tropical geometry and correspondence theorems via toric stacks
• Topology of moduli spaces of tropical curves with marked points
• On refined count of rational tropical curves
• The Newton polygon of a planar singular curve and its subdivision
• Real phase structures on matroid fans and matroid orientations
• Classification of normal toric varieties over a valuation ring of rank one
• Tropical compound matrix identities
• A guide to tropical modifications
• Dimension counts for singular rational curves via semigroups
• Toric and tropical compactifications of hyperplane complements
• A combinatorial Li-Yau inequality and rational points on curves
• The tropical $j$-invariant
• Tropical Nevanlinna theory and ultra-discrete equations
• Superforms, supercurrents, minimal manifolds and Riemannian geometry
• Some statistics about Tropical Sandpile Model
• Hyperelliptic graphs and the period mapping on outer space
• Exact Sequences in Non-Exact Categories (An Application to Semimodules)
• Welschinger invariants of small non-toric Del Pezzo surfaces
• Moduli spaces of tropical curves of higher genus with marked points and homotopy colimits
• How to repair tropicalizations of plane curves using modifications
• Dupin cyclides osculating surfaces
• Combinatorics and Genus of Tropical Intersections and Ehrhart Theory
• Tropical flag varieties
• On Tropical Linear and Integer Programs
• Tropical Theta Functions and Riemann-Roch Inequality for Tropical Abelian Surfaces
• Tropical mathematics, classical mechanics and geometry
• Number of common roots and resultant of two tropical univariate polynomials
• Idempotent and tropical mathematics. Complexity of algorithms and interval analysis
• Minkowski weights and the Grothendieck group of a toric variety
• Constructing polynomial systems with many positive solutions using tropical geometry
• Rank of tropical curves and tropical hypersurfaces
• Resultants over Commutative Idempotent Semirings
• Degenerations of amoebae and Berkovich spaces
• Tropical orbit spaces and the moduli spaces of elliptic tropical curves
• Automorphism Groups on Tropical Curves: Some Cohomology Calculations