Related papers: Tropicalization through the lens of {\L}ukasiewicz…
Tropical geometry is a relatively recent field in mathematics created as a simplified model for certain problems in algebraic geometry. We introduce the definition of abstract and planar tropical curves as well as their properties,…
A very brief introduction to tropical and idempotent mathematics is presented. Applications to classical mechanics and geometry are especially examined.
In this thesis we explore natural procedures through which topological structure can be constructed from specific semigroups. We will do this in two ways: 1) we equip the semigroup object itself with a topological structure, and 2) we find…
An algebraic setting for the validity of Pavelka style completeness for some natural expansions of \L ukasiewicz logic by new connectives and rational constants is given. This algebraic approach is based on the fact that the standard…
The main aim of this paper to show how commutative algebra is connected to topology. We give underlying topological idea of some results on completable unimodular rows.
The logico-algebraic study of Lewis's hierarchy of variably strict conditional logics has been essentially unexplored, hindering our understanding of their mathematical foundations, and the connections with other logical systems. This work…
The tropicalization of a linear space over a non-archimedean field is a tropical linear space. In this paper, we present a method for computing the tropicalization of any lattice over a valuation ring. The resulting tropical semimodule is…
In this paper we further develop the theory of geometric tropicalization due to Hacking, Keel and Tevelev and we describe tropical methods for implicitization of surfaces. More precisely, we enrich this theory with a combinatorial formula…
In this text, we merge ideas around the tropical hyperfield with the theory of ordered blueprints to give a new formulation of tropical scheme theory. The key insight is that a nonarchimedean absolute value can be considered as a morphism…
Kapranov's theorem is a foundational result in tropical geometry. It states that the set of tropicalisations of points on a hypersurface coincides precisely with the tropical variety of the tropicalisation of the defining polynomial. The…
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…
We prove that if X, X' are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly…
With the use of mathematical techniques of tropical geometry, it was shown by Mikhalkin some twenty years ago that certain Gromov-Witten invariants associated with topological quantum field theories of pseudoholomorphic maps can be computed…
This is a survey of various types of Floer theories (both in symplectic geometry and gauge theory) and relations among them.
Our objective in this project is three-fold, the first two covered in this paper. In tropical mathematics, as well as other mathematical theories involving semirings, when trying to formulate the tropical versions of classical algebraic…
Using Butz and Moerdijk's topological groupoid representation of a topos with enough points, a `syntax-semantics' duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the…
This paper surveys {\it tropical modifications}, which have already become a folklore in tropical geometry. Tropical modifications are used in tropical intersection theory, tropical Hodge theory, and in the study of singularities. They…
We study \L ukasiewicz logic enriched with a scalar multiplication with scalars taken in $[0,1]$. Its algebraic models, called {\em Riesz MV-algebras}, are, up to isomorphism, unit intervals of Riesz spaces with a strong unit endowed with…
Let $K$ be a complete, algebraically closed non-archimedean field with ring of integers $K^\circ$ and let $X$ be a $K$-variety. We associate to the data of a strictly semistable $K^\circ$-model $\mathscr X$ of $X$ plus a suitable horizontal…
We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding.…