相关论文: Tropical Geometry and its applications

The purpose of this paper is to give, on one hand, a mathematical exposition of the main topological and geometrical properties of geometric transitions, on the other hand, a quick outline of their principal applications, both in…

We continue, in this second article, the study of the the algebraic tools which play a role in tropical algebra. We especially examine here the polynomial algebras over idempotent semi-fields. this work is motivated by the development of…

This is a brief introduction to the basic concepts of topology. It includes the basic constructions, discusses separation properties, metric and pseudometric spaces, and gives some applications arising from the use of topology in computing.

Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground fields to arbitrary non-archimedean valued fields. To achieve this, we develop…

New hyperfields, that is fields in which addition is multivalued, are introduced and studied. In a separate paper these hyperfields are shown to provide a base for the tropical geometry. The main hyperfields considered here are classical…

The present text is a collection of notes about differential geometry prepared to some extent as part of tutorials about topics and applications related to tensor calculus. They can be regarded as continuation to the previous notes on…

We present tools and definitions to study abstract tropical manifolds in dimension 2, which we call simply tropical surfaces. This includes explicit descriptions of intersection numbers of 1-cycles, normal bundles to some curves and…

We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra…

A short survey on applications of algebraic geometry in topological data analysis.

These informal notes deal with some basic properties of metric spaces, especially concerning lengths of curves.

We introduce tropical complexes, as an enrichment of the dual complex of a degeneration with additional data from non-transverse intersection numbers. We define cycles, divisors, and linear equivalence on tropical complexes, analogous both…

This is a survey of the theory of real trees and their applications.

An exposition of the basic geometry of twistor integrals, intended for mathematicians.

The efficacy of using complex numbers for understanding geometric questions related to polar equations and general cycloids is demonstrated.

This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…

Numerical equivalence of algebraic cycles is defined abstractly by intersection numbers. Classically, for smooth complex proper toric varieties, the quotients by numerical equivalence with rational coefficients can be described…

These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…

The paper studies intrinsic geometry in the tropical plane. Tropical structure in the real affine $n$-space is determined by the integer tangent vectors. Tropical isomorphisms are affine transformations preserving the integer lattice of the…

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals…

We introduce the notion of resultant of two planar curves in the tropical geometry framework. We prove that the tropicalization of the algebraic resultant can be used to compute the stable intersection of two tropical plane curves. It is…