Related papers: Specialisations and Algebraically Closed Fields
In this article we use our constructions from "Enlargements of Categories" (Theory and Applications of Categories, 14:357-398) to lay down some foundations for the application of A. Robinson's nonstandard methods to modern Algebraic…
We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding…
These notes are an introduction to and an overview of the theory of algebraic surfaces over algebraically closed fields of positive characteristic. After some background in characteristic-p-geometry, we sketch the Kodaira-Enriques…
We generalize the dual notions of "expansion" and "collapse" so they can be applied to arbitrary metric spaces. We also expand the theory to allow for infinitely many such moves. Those tools are then employed to prove a variety of…
We provide a clarification of the classification of two-dimensional algebras over an arbitrary base field. Using this clarification, we determine the number of non-isomorphic two-dimensional algebras over a finite field.
We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms…
We construct the non-linear realisation of E11 and its first fundamental representation in eleven dimensions at low levels. The fields depend on the usual coordinates of space-time as well as two form and five form coordinates. We derive…
We describe the topology of singular real algebraic curves in a smooth surface. We enumerate and bound in terms of the degree the number of topological types of singular algebraic curves in the real projective plane.
We embed the geometries of the generalized $\lambda$-deformations into the framework of the Double Field Theory.
This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…
Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the `` structure sheaf algebras'' in many--in point of fact, in all--the situations of a geometrical character that…
We investigate certain arithmetic properties of field theories. In particular, we study the vacuum structure of supersymmetric gauge theories as algebraic varieties over number fields of finite characteristic. Parallel to the Plethystic…
Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field,…
We survey both old and new developments in the theory of algorithms in real algebraic geometry -- starting from effective quantifier elimination in the first order theory of reals due to Tarski and Seidenberg, to more recent algorithms for…
In order to apply nonstandard methods to modern algebraic geometry, as a first step in this paper we study the applications of nonstandard constructions to category theory. It turns out that many categorial properties are well behaved under…
We study generalization of median triangles on the plane with two complex parameters. By specialization of the parameters, we produce periodical motion of a triangle whose vertices trace each other on a common closed orbit.
The objective of this paper is to describe the structure of Zariski closed algebras, which provide a useful generalization to finite dimensional algebras in the study of representable algebras over finite fields. Our results include a…
We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.
Let $G$ be a connected reductive algebraic group with simply connected derived subgroup. Over the complex numbers there exists a local method to study the geometric properties of a point $g$ in the closure of a Jordan class of $G$ in terms…
In this note we discuss some arithmetic and geometric questions concerning self maps of projective algebraic varieties.