Related papers: Shellings of Tropical Hypersurfaces
We prove a bound conjectured by Itenberg on the Betti numbers of real algebraic hypersurfaces near non-singular tropical limits. These bounds are given in terms of the Hodge numbers of the complexification. To prove the conjecture we…
We study the tropicalization of the moduli space of algebraic spin curves, exhibit its combinatorial stratification and prove that the strata are irreducible. We construct the moduli space of tropical spin curves, prove that it is…
The prototypical examples of tropical compactifications are compactifications of complements of hyperplane arrangements, which posses a number of remarkable properties not satisfied by more general tropical compactifications of closed…
Recently, Steinberg used discrete Morse theory to give a new proof of a theorem of Symonds that the orbit space of the poset of nontrivial $p$-subgroups of a finite group is contractible. We extend Steinberg's argument in two ways, covering…
We study tropical degree bounds, stable tropical intersections, and tropical B\'ezout-type estimates through the geometry of Newton polytopes, mixed subdivisions, and lattice indices. We establish an upper bound for the tropical degree of a…
We study the question of the continuity of slices of currents and explain how it relates to several seemingly unrelated problems in tropical geometry. On the one hand, through this lens, we show that the continuity of superpotentials…
The category of exploded torus fibrations is an extension of the category of smooth manifolds in which some adiabatic limits look smooth. (For example, the limits considered in tropical geometry appear smooth, also degenerations…
We analyse the dynamics of trapped matter shells in spherically symmetric inhomogeneous \Lambda-CDM models. The investigation uses a Generalised Lema\^itre-Tolman-Bondi description with initial conditions subject to the constraints of…
The spectral problem of thin elastic shells in membrane approximation does not satisfy the classical properties of compactness and so there exists an essential spectrum. In the first part, we propose to determinate this spectrum and the…
We present extremal constructions connected with the property of simplicial collapsibility. (1) For each $d \ge 2$, there are collapsible (and shellable) simplicial $d$-complexes with only one free face. Also, there are non-evasive…
We introduce an alternative way of constructing continuous flexible tubes and tubular structures based on a discrete, semi-discrete and smooth construction of surfaces known as T-hedra in the discrete case and profile-affine surfaces in the…
First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are diffeomorphic to their degeneration called phase…
We study the tropical lines contained in smooth tropical surfaces in R^3. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however,…
While studying the continual gravitational collapse of a massive matter cloud in general relativity towards examining collapse final states, an important issue is that of whether shell-crossing singularities can develop as the collapse…
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…
Abstractly, tropical hyperelliptic curves are metric graphs that admit a two-to-one harmonic morphism to a tree. They also appear as embedded tropical curves in the plane arising from triangulations of polygons with all interior lattice…
We introduce the notion of topological hyperbolicity to characterize the largeness of the topological fundamental group of a complex variety. Inspired by the Shafarevich conjecture, we propose to study the topological hyperbolicity of…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…
Tropical implicitization means computing the tropicalization of a unirational variety from its parametrization. In the case of a hypersurface, this amounts to finding the Newton polytope of the implicit equation, without computing its…
In this paper we focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a line segment and a ray. %in \RR^{n+1}/\RR\mathbf{1}. Next we show that tropical convex…