Related papers: What can be computed in algebraic geometry?
This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel…
This paper gives an overview of some basic properties of Leibniz algebras. Some of the results were known earlier, but in the article they are accompanied by new simple proofs. Some of the results are new. The article can be viewed as a…
The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in…
We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss…
We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i.) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii.)…
In this note we discuss some arithmetic and geometric questions concerning self maps of projective algebraic varieties.
Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…
We use algebraic geometry to study matrix rigidity, and more generally, the complexity of computing a matrix-vector product, continuing a study initiated by Kumar, et. al. We (i) exhibit many non-obvious equations testing for (border)…
In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which…
Generalized geometry finds many applications in the mathematical description of some aspects of string theory. In a nutshell, it explores various structures on a generalized tangent bundle associated to a given manifold. In particular,…
We will pursue a way of building up an algebraic structure that involves, in a mathematical abstract way, the well known Grassmann variables. The problem arises when we tried to understand the grassmannian polynomial expansion on the scope…
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.
This article is an interdisciplinary review and an on-going progress report over the last few years made by myself and collaborators in certain fundamental subjects on two major theoretic branches in mathematics and theoretical physics:…
Elementary Algebraic Geometry can be described as study of zeros of polynomials with integer degrees, this idea can be naturally carried over to `polynomials' with rational degree. This paper explores affine varieties, tangent space and…
The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
This note recapitulates and expands the contents of a tutorial on the mathematical theory of algebraic effects and handlers which I gave at the Dagstuhl seminar 18172 "Algebraic effect handlers go mainstream". It is targeted roughly at the…
We discuss certain ternary algebraic structures appearing more or less naturally in various domains of theoretical and mathematical physics. Far from being exhaustive, this article is intended above all to draw attention to these algebras,…
Designing software systems for Geometric Computing applications can be a challenging task. Software engineers typically use software abstractions to hide and manage the high complexity of such systems. Without the presence of a unifying…
Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. However, it can be expensive, with worst case complexity doubly exponential in the…