Related papers: Specialisations and Algebraically Closed Fields
We survey some algebraic geometric aspects of mirror symmetry and duality in string theory. Some applications of computer algebra to algebraic geometry and string theory are shortly reviewed.
We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal…
We introduce a generalization of the Temperley--Lieb algebra. This generalization is defined by adding certain relations to the algebra of braids and ties. A specialization of this last algebra corresponds to one small Ramified Partition…
An informal discussion of how the construction problem in algebraic geometry motivates the search for formal proof methods. Also includes a brief discussion of my own progress up to now, which concerns the formalization of category theory…
We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the…
We construct and study a graded version of absolute perfectoidization for $G$-graded adic rings. As a main geometric application, we show that the absolute perfectoidization of the structure sheaf of a projective-type formal scheme admits…
Superstring compactifications have been vigorously studied for over four decades, and have flourished involving an active iterative feedback between physics and (complex) algebraic geometry. This led to an unprecedented wealth of…
Algebraic quantum field theory is an approach to relativistic quantum physics, notably the theory of elementary particles, which complements other modern developments in this field. It is particularly powerful for structural analysis but…
Given a scheme X over a field k, a generalized jet scheme parametrizes maps from Spec(A) to X, where A is a finite-dimensional, local algebra over k. We give an overview of known results concerning the dimensions of these schemes when A has…
We give a full classification, up to equivalence, of finite-dimensional graded division algebras over the field of real numbers. The grading group is any abelian group.
The geometry of algebraic curves over finite fields is a rich area of research. In previous work, the authors investigated a particular aspect of the geometry over finite fields of the classical unit circle, namely how the number of…
This article has the following aims: (1) Extend the notion of fuchsian singularities (of first kind) to base fields of arbitrary characteristic. (2) Discuss their relationship to mathematical objects of a different nature. (3) Provide a…
Leibniz algebras are certain generalization of Lie algebras. It is natural to generalize concepts in Lie algebras to Leibniz algebras and investigate whether the corresponding results still hold. In this paper we introduce the notion of…
We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.
This is a review of exceptional field theory: a generalisation of Kaluza-Klein theory that unifies the metric and $p$-form gauge field degrees of freedom of supergravity into a generalised or extended geometry, whose additional coordinates…
The authors study the method of scaling in the context of the study of automorphism groups of complex domains in multiple dimensions. Various types of scaling techniques are compared and contrasted. Applications are given in a number of…
We present an ongoing effort to implement Universal Algebra in the UniMath system. Our aim is to develop a general framework for formalizing and studying Universal Algebra in a proof assistant. By constituting a formal system for isolating…
Theory of representations of universal algebra is a natural development of the theory of universal algebra. In the book, I considered representation of universal algebra, diagram of representations and examples of representation. Morphism…
In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One…
We review the basic elements of the geometrical formalism for description of gauge fields and the theory of invariant connections, and their applications to the coset space dimensional reduction of Yang-Mills theories. We also discuss the…