Related papers: A note on gluing: a pillar of algebraic geometry

In this paper we will present an ongoing project which aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics. We will argue that this approach provides a geometric semantics for such…

This is an attempt to present axioms for Euclidean geometry, aiming at the following goals: to work with geometric notions (thus not merely identify points with pairs of numbers, giving a special status to a particular coordinate system);…

In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map…

This paper is the third in a series of papers, the aim of which is to construct algebraic geometry over metabelian Lie algebras.

The purpose of this note is to give a survey of the algebraic properties of multiplier ideals, and illustrate some of their applications to classical projective geometry.

Linear Geometry studies geometric properties which can be expressed via the notion of a line. All information about lines is encoded in a ternary relation called a line relation. A set endowed with a line relation is called a liner. So,…

This note surveys how the exterior algebra and deformations or quotients of it, gives rise to centrally important notions in five domains of mathematics: Combinatorics, Topology, Lie theory, Mathematical physics, and Algebraic geometry.

We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the…

In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along…

Since the time when the first optical instruments have been invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way to formalize it in mathematics is…

Using log-geometry, we construct a model for the configuration category of a smooth algebraic variety. As an application, we prove the formality of certain configuration spaces.

We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually the integers, rationals, or…

The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the…

Three types of geometric structure---grid triangulations, rectangular subdivisions, and orthogonal polyhedra---can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an…

We consider relationships between cubic algebras and implication algebras. We first exhibit a functorial construction of a cubic algebra from an implication algebra. Then we consider an collapse of a cubic algebra to an implication algebra…

This note is the sequel of "Geometric structures as variational objects, I." It generalizes the main result and perspectives of that work to a class of geometric structures that includes integrable almost-complex structures.

Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and…

This paper is a sequel to [3]. We formulate a natural algebraic geometry conjecture, give some of its number theoretic and analytical consequences, and show that those can be used to get further advances in wave turbulence theory.

The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. This paper studies analogical proportions in the boolean domain consisting of two elements 0 and 1…

This is a first step guide to the theory of cluster algebras. We especially focus on basic notions, techniques, and results concerning seeds, cluster patterns, and cluster algebras.