Related papers: Three lectures on tropical algebra
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 notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational perspective. Our results include a convex hull…
We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing…
We develop the rudiments of a finite-dimensional representation theory of groups over idempotent semifields by considering linear actions on tropical linear spaces. This can be considered a tropical representation theory, a characteristic…
The purpose of this article is to investigate triangularization and simultaneous triangularization of matrices over max algebras using graph theoretic methods. We establish a connection between commutators and commutants with simultaneous…
Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial…
We construct a general framework for tropical differential equations based on idempotent semirings and an idempotent version of differential algebra. Over a differential ring equipped with a non-archimedean norm enhanced with additional…
In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the…
We propose to take a look at a new approach to the study of integral polyhedra. The main idea is to give an integral representation, or matrix model representation, for the key combinatorial characteristics of integral polytopes. Based on…
This is an expository article on the adic tropicalization of algebraic varieties. We outline joint work with Sam Payne in which we put a topology and structure sheaf of local topological rings on the exploded tropicalization. The resulting…
Let $I$ be an ideal of the ring of Laurent polynomials $K[x_1^{\pm1},\ldots,x_n^{\pm1}]$ with coefficients in a real-valued field $(K,v)$. The fundamental theorem of tropical algebraic geometry states the equality…
Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical…
This basic introduction to tropical geometry is hopefully accessible to a first years student in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking.…
These are notes from the lectures I gave at the Oberwolfach seminar `Tensor Triangular Geometry and Interactions' which was held in October 2025. The aim of these notes is to give an introduction to tensor triangular geometry, for both…
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.
We define tropical analogues of the notions of linear space and Plucker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated…
We introduce algebraic structures on the polyvector fields of an algebraic torus that serve to compute multiplicities in tropical and log Gromov-Witten theory while also connecting to the mirror symmetry dual deformation theory of complex…
This is a write-up of the author's invited talk at the Eighth International Congress of Chinese Mathematicians (ICCM) held at Beijing in June 2019. We give a survey on joint works with Naichung Conan Leung and Ziming Nikolas Ma where we…
In this paper we further develop the theory of matrices over the extended tropical semiring. Introducing a notion of tropical linear dependence allows for a natural definition of matrix rank in a sense that coincides with the notions of…
This work uncovers the tropical analogue for measured laminations of the convex hull construction of decorated Teichmueller theory, namely, it is a study in coordinates of geometric degeneration to a point of Thurston's boundary for…