Related papers: Wall-crossing and recursion formulae for tropical …
We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds, which we briefly…
Extremality and irreducibility constitute fundamental concepts in mathematics, particularly within tropical geometry. While extremal decomposition is typically computationally hard, this article presents a fast algorithm for identifying the…
A polynomial complexity algorithm is designed which tests whether a point belongs to a given tropical linear variety.
Recently, the first and third author proved a correspondence theorem which recovers the Levine-Welschinger invariants of toric del Pezzo surfaces as a count of tropical curves weighted with arithmetic multiplicities. In this paper, we study…
This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories,…
For certain tropical quartic curves the existing techniques could not predict the lifting behavior of their bitangents over the real numbers. We close this gap by using patchworking techniques. Further, this paper provides an analysis of…
We study higher rank Donaldson-Thomas invariants of a Calabi-Yau 3-fold using Joyce-Song's wall-crossing formula. We construct quivers whose counting invariants coincide with the Donaldson-Thomas invariants. As a corollary, we prove the…
We establish a new notion of tropical convexity for signed tropical numbers. We provide several equivalent descriptions involving balance relations and intersections of open halfspaces as well as the image of a union of polytopes over…
The main result of this paper is a proof using real analysis of the monotonicity of the topological entropy for the family of quadratic maps, sometimes called Milnor's Monotonicity Conjecture. In contrast, the existing proofs rely in one…
Motivated by results for the HCIZ integral in Part I of this paper, we study the structure of monotone Hurwitz numbers, which are a desymmetrized version of classical Hurwitz numbers. We prove a number of results for monotone Hurwitz…
Consider a pair consisting of an abstract tropical curve and an effective divisor from the linear system associated to $k$ times the canonical divisor for $k \in \mathbb{Z}_{\geq 1}$. In this article we give a purely combinatorial criterion…
In this thesis we study the relation between scattering diagrams and deformations of holomorphic pairs, building on a recent work of Chan--Conan Leung--Ma. The new feature is the extended tropical vertex group where the scattering diagrams…
This paper presents numerical solutions and idealized analytical solutions of axisymmetric, $f$-plane models of the tropical cyclone boundary layer. In the numerical model, the boundary layer radial and tangential flow is forced by a…
We present a simple and elementary procedure to sketch the tropical conic given by a degree--two homogeneous tropical polynomial. These conics are trees of a very particular kind. Given such a tree, we explain how to compute a defining…
We consider symbolic flows over finite alphabets and study certain kinds of repetitions in these sequences. Positive and negative results for the existence of such repetitions are given for codings of interval exchange transformations and…
We reconcile the discrepancy between the complex and tropical counts of some enumerative problems reducing to positive characteristic. Each problem that we consider suggests a prime with special behaviour. Modulo this prime, the solutions…
We study the structures of ordinary simple Hurwitz numbers and monotone Hurwitz numbers with varying genus. More precisely, we prove that when the ramification type is fixed and the genus is treated as a variable, the connected monotone…
We tropicalize the rational map that takes triples of points in the projective plane to the plane of quadrics passing through these points. The image of its tropicalization is contained in the tropicalization of its image. We identify these…
We provide a wall-crossing framework for operational enumerative invariants of equivariant 3-Calabi--Yau categories arising from virtual cycles. The strategy follows ideas of Joyce's ``universal'' wall-crossing framework arXiv:2111.04694,…
We express the reduction types of Picard curves in terms of tropical invariants associated to binary quintics. We also give a general framework for tropical invariants associated to group actions on arbitrary varieties. The problem of…