Related papers: A tropical Krull-Schmidt theorem
The main purpose of this paper is to study the vector groupoids. This is an algebraic structure which combines the concepts of Brandt groupoid and vector space such that these are compatible.
We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted…
We develop a new framework of relative algebroids to address existence and classification problems of geometric structures subject to partial differential equations.
We develop a theory of toroidal vertex algebras and their modules, and we give a conceptual construction of toroidal vertex algebras and their modules. As an application, we associate toroidal vertex algebras and their modules to toroidal…
We survey some results that provide different versions of classical results through different summability methods. Specifically, in order to adapt such classical results, we analyze which properties should satisfy the summability methods.…
We introduce tropical Kummer quartic surfaces in tropical projective $3$-space as the images of certain principally polarized tropical abelian surfaces under tropical theta functions of second order. We study some of their properties,…
We study the geometry of tropical extensions of hyperfields, including the ordinary, signed and complex tropical hyperfields. We introduce the framework of 'enriched valuations' as hyperfield homomorphisms to tropical extensions, and show…
PhD dissertation consists in three lines of investigation involving rational elliptic surfaces, namely 1) a study of conic bundles on these surfaces; 2) an investigation of the possible intersection numbers of two sections and 3) a theorem…
We present projective versions of the center point theorem and Tverberg's theorem, interpolating between the original and the so-called "dual" center point and Tverberg theorems. Furthermore we give a common generalization of these and many…
We consider the (direct sum over all $n$ of the) $K$-theory of the semi-nilpotent commuting variety of $\mathfrak{gl}_n$, and describe its convolution algebra structure in two ways: the first as an explicit shuffle algebra (i.e. a…
We introduce and study categorical realizations of quivers. This construction generalizes comma categories and includes representations of quivers on categories, twisted representations of quivers and bilinear pairings as special cases. We…
We study the notion of singular tropical hypersurfaces of any dimension. We characterize the singular points in terms of tropical Euler derivatives and we give an algorithm to compute all singular points. We also describe non-transversal…
We construct algebraic curves in abelian surfaces starting from tropical curves in real tori. We give a necessary and sufficient condition for a tropical curve in a real torus to be realizable by an algebraic curve in an abelian surface.…
We prove the conjecture that higher Verlinde categories are geometrically reductive. This is one of the two properties required in order for recent results on algebraic geometry in tensor categories to apply to these categories. We also…
As a new concept tropical halfspaces are introduced to the (linear algebraic) geometry of the tropical semiring (R,min,+). This yields exterior descriptions of the tropical polytopes that were recently studied by Develin and Sturmfels in a…
The aim of this paper is to present the main constructions of the substructures of an almost groupoid and to discuss their basic properties. The definitions and properties concerning these new algebraic constructions extend to almost…
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,…
We algebraically compute all possible sectional curvature values for canonical algebraic curvature tensors, and use this result to give a method for constructing general sectional curvature bounds. We use a well-known method to…
We first show that increasing trees are in bijection with set compositions, extending simultaneously a recent result on trees due to Tonks and a classical result on increasing binary trees. We then consider algebraic structures on the…
This paper investigates the interplay between algebraic structure, topology, and differentiability in Clifford semigroups. The study is developed along three main themes. First, in the compact Hausdorff setting, we provide an explicit…