Tropical Geometry and its applications

By G. Mikhalkin. 2006

370 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/b485e98bca98ee1427e7d8a0fe9eec534f54c931

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Computational Theory and Mathematics

Citations in the corpus

Formes différentielles réelles et courants sur les espaces de Berkovich (2012, positive): “Introduit initialement dans le cas où k est le corps des nombres complexes et X est une variété algébrique (voir [6, 56, 47]), ce point de vue a ensuite été développé dans le cas ultramétrique (citons notamment [10, 33, 39]).”
Tropical hyperelliptic curves (2011, methodology): “This space was considered by Mikhalkin and Zharkov in [Mik06] and [MZ07], in a more limited setting (i.”
Tropical intersection theory from toric varieties (2009, background): “Define the Mikhalkin intersection product C1 • C2 [14] as follows: subdivide C1 and C2 such that their intersection C1 ∩C2 is a subcomplex of each; for each (k+ l)−ndimensional cell τ of C1 ∩ C2; let ∆1 = StarC1(τ), ∆2 = StarC2(τ); pick a complete fan ∆ that contains refinements of ∆1 and ∆2 as subc”
Labeled floor diagrams for plane curves (2009, background): “Let us review the basic notions of tropical curves, both abstract and parametrized; see [21, 27, 28] for further details.”
Analytification is the limit of all tropicalizations (2008, methodology): “Here we use an extended tropicalization map from Am(K) to Rm, following well-known ideas of Mikhalkin [Mik06], Speyer and Sturmfels [SS04b], and others.”
Brief introduction to tropical geometry (2015, methodology): “We direct the reader to [Mik06] or [MZ14] for more details.”
A Field of Generalised Puiseux Series for Tropical Geometry (2007, neutral): “(12) fi,ν = n ∑ j=i f j ,ν−1 · t j ·ωn−1−αν−1 · ( j i ) ·c j−i ν−1, where for eachν ∈ N αν = val( f0,ν)> 0 and”
A Tropical Toolkit (2006, methodology): “into Torus Orbits 35 References 36 1 Introduction Tropical Geometry is an exciting new ﬁeld of mathematics arising out of computer science. In the mathematical realm, it has been studied by Mikhalkin [24], Speyer Preprint submitted to Elsevier 4 April 2008 [33], the Sturmfels school [30], Itenberg, ”
Superforms, tropical cohomology, and Poincaré duality (2015, background): “ing definition is a generalization of the definition of tropical spaces given in [19; 21; 6].”
TORELLI THEOREM FOR GRAPHS AND TROPICAL CURVES (2009, methodology): “ turn to another, recently discovered aspect of the analogy between graphs and curves, that is, the tight connection between tropical curves and graphs. By results of G. Mikhalkin and I. Zharkov, see [Mik06] and [MZ07], there exists a natural bijection between the set of tropical equivalence classes”
Super currents and tropical geometry (2010, background): “2 The intersection theory developed here seems to fit well with the intersection theory for tropical geometry considered in [10].”