Related papers: Tropicalisation for Topologists
A toric degeneration in algebraic geometry is a process where a given projective variety is being degenerated into a toric one. Then one can obtain information about the original variety via analyzing the toric one, which is a much easier…
We survey some results on real rational surfaces focused on their topology and their birational geometry.
On objects of a triangulated category with a stability condition, we construct a topology.
This paper provides an overview of modern digital geometry and topology through mathematical principles, algorithms, and measurements. It also covers recent developments in the applications of digital geometry and topology including image…
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 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…
In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian…
A new methodological approach for the study of topology for shapes made of arrangements of lines, planes or solids is presented. Topologies for shapes are traditionally built on the classical theory of point-sets. In this paper, topologies…
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…
We study tropical line arrangements associated to a three-regular graph $G$ that we refer to as \emph{tropical graph curves}. Roughly speaking, the tropical graph curve associated to $G$, whose genus is $g$, is an arrangement of $2g-2$…
The goal of this survey is to give a list of resent results about topology of manifolds admitting different metrics with the same geodesics. We emphasize the role of the theory of integrable systems in obtaining these results.
Let $X$ be an algebraic variety and let $S$ be a tropical variety associated to $X$. We study the tropicalization map from the moduli space of stable maps into $X$ to the moduli space of tropical curves in $S$. We prove that it is a…
In this paper, we give a survey of a geometrical theory of Jacobi forms of higher degree. And we present some geometric results and discuss some geometric problems to be investigated in the future.
We investigate the tree gonality of a genus-$g$ metric graph, defined as the minimum degree of a tropical morphism from any tropical modification of the metric graph to a metric tree. We give a combinatorial constructive proof that this…
This paper surveys some results and methods in topological transformation groups.
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…
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.
A graph profile records all possible densities of a fixed finite set of graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known, and little is known about hypergraph…
We present the topological classification of real parts of real regular elliptic surfaces with a real section.
This paper presents a gentle and informal introduction to the Skorokhod topologies. Focus is on motivating examples and concepts.