Related papers: Un peu de geometrie tropicale
We study a tropical analogue of the projective dual variety of a hypersurface. When $X$ is a curve in $\mathbb{P}^2$ or a surface in $\mathbb{P}^3$, we provide an explicit description of $\text{Trop}(X^*)$ in terms of $\text{Trop}(X)$, as…
We define Galois coverings on tropical curves for which a Galois correspondence and a universal mapping property hold.
This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, arrangements of finitely many tropical halfspaces can be…
We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.
Exploiting a connection between amoebas and tropical curves, we devise a method for computing tropical curves using numerical algebraic geometry and give an implementation. As an application, we use this technique to compute Newton polygons…
In this paper we explain four viewpoints on the local tropicalization of formal subgerms of toric germs, which is a local analog of the global tropicalization of subvarieties of algebraic tori. We start by illustrating some of those…
We study the tropicalizations of Severi varieties, which we call tropical Severi varieties. In this paper, we give a partial answer to the following question, ``describe the tropical Severi varieties explicitly.'' We obtain a description of…
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…
We describe a framework to construct tropical moduli spaces of rational stable maps to a smooth tropical hypersurface or curve. These moduli spaces will be tropical cycles of the expected dimension, corresponding to virtual fundamental…
We propose to study the tropical geometry specifically arising from convergent Hahn series in multiple indeterminates. One application is a new view on stable intersections of tropical hypersurfaces. Another one is perturbations of rank one…
The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized (max, +),…
This is a first graduate course in algebraic geometry. It aims to give the student a lift up into the subject at the research level, with lots of interesting topics taken from the classification of surfaces, and a human-oriented discussion…
We study pairs of mutually orthogonal normal matrices with respect to tropical multiplication. Minimal orthogonal pairs are characterized. The diameter and girth of three graphs arising from the orthogonality equivalence relation are…
This paper will appear in the Proceedings of the 1995 Santa Cruz Summer Institute. The paper is a survey of recent developments in the theory of toric varieties, including new constructions of toric varieties and relations to symplectic…
These notes are based on an introductory minicourse on Poisson geometry given at CRM, Barcelona, in July 2022. They mostly contain foundational material, including motivating questions and key examples of Poisson structures, and highlight…
We present an overview of the differential geometry of curves and surfaces using examples from soft matter as illustrations. The presentation requires a background only in vector calculus and is otherwise self-contained.
This paper presents a brief but comprehensive introduction to certain mathematical techniques in General Relativity. Familiar mathematical procedures are investigated taking into account the complications of introducing a non trivial…
Analogously as in classical algebraic geometry, linear pencils of tropical plane curves are parameterized by tropical lines in a coefficient space. A special example of such a linear pencil is the set of tropical plane curves with an…
In this paper we initiate the study of tropical Voronoi diagrams. We start out with investigating bisectors of finitely many points with respect to arbitrary polyhedral norms. For this more general scenario we show that bisectors of three…
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:…