Related papers: On Bands and Limit Theorems in Tropical Geometry
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 prove a fundamental theorem for tropical partial differential equations, analogous to the fundamental theorem of tropical geometry in this context. We extend results from Aroca et al., Falkensteiner et al. and from Fink and Toghani for…
In this work, we argue that the $\alpha'\to 0$ limit of closed string theory scattering amplitudes is a tropical limit. The motivation is to develop a technology to systematize the extraction of Feynman graphs from string theory amplitudes…
In this paper we propose a general functorial definition of the operation of \emph{local tropicalization} in commutative algebra. Let $R$ be a commutative ring, $\Gamma$ a finitely generated subsemigroup of a lattice, $\gamma : \Gamma…
Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate…
We consider elliptic partial differential equations with diffusion coefficients that depend affinely on countably many parameters. We study the summability properties of polynomial expansions of the function mapping parameter values to…
In this paper, we propose an unfitted Nitsche's method to compute the band structures of phononic crystal with impurities of general geometry. The proposed method does not require the background mesh to fit the interfaces of impurities, and…
We introduce Liu algebras as Banach algebras which are 'locally affinoid', and define non-Archimedean Stein algebras as suitable inverse limits of these. We show that this gives rise to a complete functorial characterisation of…
We introduce tropical dual numbers as an extension of tropical semiring. By this innovation, one can work with honest ideals, instead of congruences, and recover the Euclidean topology on affine tropical spaces similar to Zariski's approach…
The finite strain theory is reformulated in the frame of the Tangential Differential Calculus (TDC) resulting in a unification in a threefold sense. Firstly, ropes, membranes and three-dimensional continua are treated with one set of…
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…
We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also…
We present two effective tools for computing the positive tropicalization of algebraic varieties. First, we outline conditions under which the initial ideal can be used to compute the positive tropicalization, offering a real analogue to…
We show that in the decoupling limit of an F-theory compactification, the internal directions of the seven-branes must wrap a non-commutative four-cycle S. We introduce a general method for obtaining fuzzy geometric spaces via toric…
The transfer-matrix methodology is used to solve linear systems of differential equations, such as those that arise when solving Schr\"odinger's equation, in situations where the solutions of interest are in the continuous part of the…
Topological principles constitute at present an integral component of condensed matter physics, permeating the modern characterization of electronic states while also guiding materials design. In this brief Perspective, I highlight three…
The main goal of this review is to compare different approaches to constructing geometry associated with a Hecke type braiding (in particular, with that related to the quantum group U_q(sl(n))). We make an emphasis on affine braided…
We use topological methods to study various semicontinuity properties of spectra of singular points of plane algebraic curves and of polynomials in two variables at infinity. Using Seifert forms and the Tristram--Levine signatures of links,…
We investigate the possibility of exactly flat non-trivial Chern bands in tight binding models with local (strictly short-ranged) hopping parameters. We demonstrate that while any two of three criteria can be simultaneously realized…
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…