Related papers: Quasilinear tropical compactifications
We introduce tropical analogues of the notion of volume of polytopes, leading to a tropical version of the (discrete) classical isoperimetric inequality. The planar case is elementary, but a higher-dimensional generalization leads to an…
An unconstrained optimization problem is formulated in terms of tropical mathematics to minimize a functional that is defined on a vector set by a matrix and calculated through multiplicative conjugate transposition. For some particular…
Directional notions in topology and analysis naturally lead to nonsymmetric structures such as quasi-metrics, quasi-uniformities, and modular spaces. In these settings, classical notions of connectedness and completion based on symmetric…
Following our previous work, we develop an algorithm to compute a presentation of the fundamental group of certain partial compactifications of the complement of a complex arrangement of lines in the projective plane. It applies, in…
We study the compactification of the $(2,0)$ and type-II little-string theories on $S^1$, $T^2$ and $T^3$ with an R-symmetry twist that preserves half the supersymmetry. We argue that it produces the same moduli spaces of vacua as…
The goal of this paper is to make a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. The connection allows us to study important and difficult questions in these areas:…
We show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich's sense by constructing a continuous section to the tropicalization map. Our main tool is an explicit…
We study the moduli space of $d$-dimensional linear subspaces contained in a fixed $(d+1)$-dimensional linear variety $X$, and its tropicalization. We prove that these moduli spaces are linear subspaces themselves, and thus their…
We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either…
The tropical semifield, i.e., the real numbers enhanced by the operations of addition and maximum, serves as a base of tropical mathematics. Addition is an abelian group operation, whereas the maximum defines an idempotent semigroup…
Atomic-resolution electron microscope images show that a quasicrystal is a quasiperiodic packing of clusters. The outer atomic shells of multi-shell clusters occuring in quasicrystals are highly symmetric and rather robust, but some…
We investigate (quasi)varieties of lattices with complementation, i.e., complemented lattices equipped with a fixed complementation as a unary operation. We focus on subclasses satisfying additional conditions, such as the quasi-identity…
The set of quasipositive surfaces is closed under incompressible inclusion. We prove that the induced order on fibre surfaces of positive braid links is almost a well-quasi-order. When restricting to quasipositive surfaces containing a…
In this paper we construct a compactification for the parameter space of convex projective structures on a fixed n-manifold M. This parameter space is a closed semi-algebraic subset of the variety of characters of representations of the…
This is a survey on an analogue of tropical convexity developed over the max-min semiring, starting with the descriptions of max-min segments, semispaces, hyperplanes and an account of separation and non-separation results based on…
In this work we generalize the classical notion of a (compact) twistor line in the period domain of compact complex tori. We introduce two new types of lines, which are non-compact analytic curves in the period domain of complex tori. We…
Tilings and point sets arising from substitutions are classical mathematical models of quasicrystals. Their hierarchical structure allows one to obtain concrete answers regarding spectral questions tied to the underlying measures and…
In this work we investigate a largely unexplored non-geometric corner of the string landscape: the quasicrystalline orbifolds. These exist at special points of the Narain moduli leading to frozen moduli and large quantum symmetries. Here we…
In this paper we give an interpretation to the boundary points of the compactification of the parameter space of convex projective structures on an n-manifold M. These spaces are closed semi-algebraic subsets of the variety of characters of…
Let $\Omega$ be the complement of a connected, essential hyperplane arrangement. We prove that every dominant endomorphism of $\Omega$ extends to an endomorphism of the tropical compactification $X$ of $\Omega$ associated to the Bergman fan…