Related papers: Wave fronts and caustics in the tropical plane
Metric graphs are important models for capturing the structure of complex data across various domains. While much effort has been devoted to extracting geometric and topological features from graph data, computational aspects of metric…
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…
We show that an embedding in Euclidean space based on tropical geometry generates stable sufficient statistics for barcodes. In topological data analysis, barcodes are multiscale summaries of algebraic topological characteristics that…
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…
We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative…
We consider a toric degeneration of Calabi--Yau complete intersections of Batyrev--Borisov in the Gross--Siebert program. One can associate two types of tropical spaces with it. One is a tropical variety obtained by tropicalization. The…
Extreme value statistics is the max analogue of classical statistics, while tropical geometry is the max analogue of classical geometry. In this paper, we review recent work where insights from tropical geometry were used to develop new,…
In this paper, we give a tropical method for computing Gromov-Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds which admit toric degenerations to toric Fano varieties with singularities…
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 study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all \ ultrametrics. The ${\rm CAT}(0)$-metric of Billera-Holmes-Vogtman arises from the…
Tropical refined invariants of toric surfaces constitute a fascinating interpolation between real and complex enumerative geometries via tropical geometry. They were originally introduced by Block and G\"ottsche, and further extended by…
We present an asymptotic and numerical study of the evolution of an incoming wavefield which has a caustic close to a curve with an inflection point. Our results reveal the emergence of a wavefield which resembles that of a shadow boundary…
We show that the commutator relations in the refined tropical vertex group can be expressed via the enumeration of suitable real rational curves in toric surfaces.
We define the tropical Tevelev degrees, $\mathsf{Tev}_g^{trop}$, as the degree of a natural finite morphism between certain tropical moduli spaces, in analogy to the algebraic case. We develop an explicit combinatorial construction that…
For any stable curve $f$ in an exploded manifold, this paper constructs a family of curves $\hat f$ with universal tropical structure which contains $f$. Such a family has the property that any other family of curves containing $f$ is…
This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's…
We study the stationary descendant Gromov-Witten theory of toric surfaces by combining and extending a range of techniques - tropical curves, floor diagrams, and Fock spaces. A correspondence theorem is established between tropical curves…
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…
We revisit the representation theory in type $A$used previously to establish that the dissimilarity vectors of phylogenetic trees are points on the tropical Grassmannian variety. We use a different version of this construction to show that…
We continue our investigation of tropical branes by exploring the tropicalization of topological sigma models with boundaries. We show that the tropical limit naturally decomposes conventional A-branes into two distinct classes: tropical…