相关论文: Elements of Nonstandard Algebraic Geometry

We give a survey on Auslander-Gorenstein algebras with a focus on finite-dimensional algebras. We put an emphasis on recent classification results for special classes of algebras and the newly discovered interactions of the Auslander-Reiten…

A variety of associative algebras is called nonmatrix if it does not contain the algebra of 2 x 2 matrices over the given field. Nonmatrix varieties were introduced and studied by V.N.Latyshev in relation with the Specht problem. Some…

We study equations over boolean algebras with distinguished elements. We prove the criteria, when a boolean algebra is equationally Noetherian, weakly equationally Noetherian, $\mathbf{q}_\omega$-compact or $\mathbf{u}_\omega$-compact. Also…

The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…

We investigate equivariant birational geometry of rational surfaces and threefolds from the perspective of derived categories.

This paper focuses on polynomial dynamical systems over finite fields. These systems appear in a variety of contexts, in computer science, engineering, and computational biology, for instance as models of intracellular biochemical networks.…

We study varieties defined by parameterizing polynomials of derivatives through a computational algebro-geometric approach, especially relying on Combinatorial Nullstellensatz and Noether normalization. We establish that these polynomials…

Shape is an important physical property of natural and manmade 3D objects that characterizes their external appearances. Understanding differences between shapes and modeling the variability within and across shape classes, hereinafter…

Let A be a finitely generated associative algebra over an algebraically closed field. We characterize the finite dimensional modules over A whose orbit closures are regular varieties.

We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes, due to W. Messing and the author (arXiv:math.AG/0106083), and give a more direct derivation of the associated cocycle equations.…

We survey on algebraically elliptic varieties in the sense of Gromov.

A one parameter set of noncommutative complex algebras is given. These may be considered deformation quantisation algebras. The commutative limit of these algebras correspond to the algebra of polynomial functions over a manifold or…

We introduce and investigate the concept of Stratified Algebra, a new algebraic framework equipped with a layer-based structure on a vector space. We formalize a set of axioms governing intra-layer and inter-layer interactions, study their…

Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…

We introduce an approach to the categorification of rings, via the notion of distributive categories with negative objects, and use it to lay down categorical foundations for the study of super, quantum and non-commutative combinatorics.…

We construct examples of non-schematic algebraic spaces that become schemes after finite ground field extensions.

We survey the field of nonparametric inference under shape constraints, providing a historical overview and a perspective on its current state. An outlook and some open problems offer thoughts on future directions.

In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers.

The goal of this present manuscript is to introduce the reader to the nonstandard method and to provide an overview of its most prominent applications in Ramsey theory and combinatorial number theory.

We construct supersymmetric K field theories (i.e., theories with a non-standard kinetic term) in 1+1 and 2+1 dimensions such that the bosonic sector just consists of a nonstandard kinetic term plus a potential. Further, we study the…