Related papers: Brackets and Projective Geometry in Macaulay2
We describe the Macaulay2 package TateOnProducts and its capabilities, which include computing cohomology tables and Beilinson monads of sheaves on products of projective spaces and the derived category pushForward of a sheaf under a…
We review origins and main properties of the most important bracket operations appearing canonically in differential geometry and mathematical physics in the classical, as well as the supergeometric setting. The review is supplemented by a…
We survey recent developments in the Birational Anabelian Geometry program aimed at the reconstruction of function fields of algebraic varieties over algebraically closed fields from pieces of their absolute Galois groups.
This is an overview of Erlangen Programme at Large. Study of objects and properties, which are invariant under a group action, is very fruitful far beyond the traditional geometry. In this paper we demonstrate this on the example of the…
This note describes a \emph{Macaulay2} package for computations in prime characteristic commutative algebra. This includes Frobenius powers and roots, $p^{-e}$-linear and $p^{e}$-linear maps, singularities defined in terms of these maps,…
We outline a program in the area of formalization of mathematics to automate theorem proving in algebra and algebraic geometry. We propose a construction of a dictionary between automated theorem provers and (La)TeX exploiting syntactic…
The Macaulay2 package PHCpack.m2 provides an interface to PHCpack, a general-purpose polynomial system solver that uses homotopy continuation. The main method is a numerical blackbox solver which is implemented for all Laurent systems. The…
A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet…
We employ techniques of machine-learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether AI can "learn" algebraic structures. Using finite groups and finite rings as a concrete playground, we…
We give an overview of a Macaulay2 package for computing with the multigraded BGG correspondence. This software builds on the package BGG due to Abo-Decker-Eisenbud-Schreyer-Smith-Stillman, which concerns the standard graded BGG…
A criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. Several novel examples of plane curves with two inner Galois points as an application are described.
Bijections between invariants associated to indecomposable projective modules over some suitable Brauer configuration algebras and invariants associated to solutions of the Kronecker problem and the four subspace problem are used to…
We introduce braided Lie bialgebras as the infinitesimal version of braided groups. They are Lie algebras and Lie coalgebras with the coboundary of the Lie cobracket an infinitesimal braiding. We provide theorems of transmutation, Lie…
In this article we show how to compute a matrix representation and the implicit equation by means of the method developed in [Botbol: arXiv:1007.3437], using the computer algebra system Macaulay2 \cite{M2}. As it is probably the most…
This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis, their properties depend on…
We survey the application of computer algebra in the context of gravitational theories. After some general remarks, we show of how to check the second Bianchi-identity by means of the Reduce package Excalc. Subsequently we list some…
We give illustrative examples of how the computer algebra system OSCAR can support research in commutative algebra and algebraic geometry. We start with a thorough introduction to Groebner basis techniques, with particular emphasis on the…
The Boolean algebra of regular closed sets is prominent in topology, particularly as a dual for the Stone-Cech compactification. This algebra is also central for the theory of geometric computation, as a representation for combinatorial…
The first part of this paper discusses general procedures for finding numerical approximations to distinguished Kahler metrics, such as Calabi-Yau metrics, on complex projective manifolds. These procedures are closely related to ideas from…
We study Lie-Rinehart algebra structures in the framework provided by a duality pairing of modules over a unital commutative associative algebra. Thus, we construct examples of Lie brackets corresponding to a fixed anchor map whose image is…