Related papers: Tropical Mathematics
Tropical algebraic geometry is an active new field of mathematics that establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones. This expository paper gives an introduction…
A very brief introduction to tropical and idempotent mathematics is presented. Applications to classical mechanics and geometry are especially examined.
This document is a slightly expanded version of a series of talks given by J. Giansiracusa at the workshop `Geometry over semirings' at Universitat Aut\`{o}noma de Barcelona in July 2025. In the first lecture we introduce tropical…
These condensed notes treat some basic notions in Tropical Geometry (varieties, cycles, modifications, equivalence). These topics are to be extended, illustrated and included to the upcoming book project…
This short article, which appeared in the MSRI Fall 2004 newletter, is intended for a general mathematical audience. It describes some mathematical themes from the Winter 2004 program at MSRI in topological aspects of real algebraic…
The term "tropical convexity" was coined by Develin and Sturmfels who published a landmark paper with that title in 2004. However, the topic has much older roots and is deeply connected to linear and combinatorial optimization and other…
In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…
A brief introduction to tropical and idempotent mathematics (with an emphasys on idempotent functional analysis) is presented. Applications to classical mechanics and geometry are especially examined.
Tropical mathematics is used to establish a correspondence between certain microscopic and macroscopic objects in statistical models. Tropical algebra gives a common framework for macrosystems (subsets) and their elementary constituents…
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.
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…
This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial…
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…
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…
In recent years a series of remarkable advances in tropical geometry and in non-archimedean geometry have brought new insights to the moduli theory of algebraic curves and their Jacobians. The goal of this survey, an expanded version of my…
The paper consists of lecture notes for a mini-course given by the authors at the G\"okova Geometry \& Topology conference in May 2014. We start the exposition with tropical curves in the plane and their applications to problems in…
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…
We study Alexandrov curvature in the tropical projective torus with respect to the tropical metric, which has been useful in various statistical analyses, particularly in phylogenomics. Alexandrov curvature is a generalization of classical…
In this survey, we discuss linear series on tropical curves and their relation to classical algebraic geometry, describe the main techniques of the subject, and survey some of the recent major developments in the field, with an emphasis on…
These are notes to accompany four lectures that I gave at the School on Additive Combinatorics, held in Montreal, Quebec between March 30th and April 5th 2006. My aim is to introduce ``quadratic fourier analysis'' in so far as we understand…