Related papers: Wave fronts and caustics in the tropical plane
Tropical caustic of a convex domain on the plane is a canonically associated tropical analytic curve inside the domain. In this note we give a graphical proof for the classification of its intermediate vertices, implying in particular that…
The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it…
The goal of this paper is to make a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. The connection allows us to study important and difficult questions in these areas:…
A tropical expansion is a degeneration of a toroidal embedding, induced by a polyhedral subdivision of its tropicalisation. Each irreducible component of a tropical expansion admits a collapsing map down to a stratum of the original…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically…
This is a survey article written for the Jahresberichte der DMV. Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and…
Tropical manifolds are polyhedral complexes enhanced with certain kind of affine structure. This structure manifests itself through a particular cohomology class which we call the eigenwave of a tropical manifold. Other wave classes of…
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on…
Abstractly, tropical hyperelliptic curves are metric graphs that admit a two-to-one harmonic morphism to a tree. They also appear as embedded tropical curves in the plane arising from triangulations of polygons with all interior lattice…
In classical geometry, a linear space is a space that is closed under linear combinations. In tropical geometry, it has long been a consensus that tropical varieties defined by valuated matroids are the tropical analogue of linear spaces.…
Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. They are also objects of interest in pure mathematics, such as algebraic geometry and combinatorics, due to their discrete geometry.…
We present applications of tropical geometry to some integrable piecewise-linear maps, based on the lecture given by one of the authors (R. I.) at the workshop "Tropical Geometry and Integrable Systems" (University of Glasgow, July 2011),…
We study the tropical version of the contraction morphism $\mathcal{T}$ between moduli spaces of stable and pseudostable curves. By promoting $\mathcal{T}$ to a logarithmic morphism, we obtain a piecewise linear function between the…
Tropical convective clouds evolve over a wide range of temporal and spatial scales, and this makes them difficult to simulate numerically. Here, we propose that their statistical properties can be derived within a simplified…
The tropical convex hull of a finite set of points in tropical projective space has a natural structure of a cellular free resolution. Therefore, methods from computational commutative algebra can be used to compute tropical convex hulls.…
Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. The purpose of this paper is to advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the…
Tropical geometry is a relatively recent field in mathematics created as a simplified model for certain problems in algebraic geometry. We introduce the definition of abstract and planar tropical curves as well as their properties,…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…