Related papers: Adelic Geometry and Polarity
The topics of Convexity and Concavity and Envelopes are central in Complex Analysis and extensively investigated. The aim of this paper is to find a possible counterpart in Algebraic Geometry. The article presents preliminary results on…
This is a brief review article of various applications of non-Archimedean geometry, p-adic numbers and adeles in modern mathematical physics.
We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex…
Polarity is a fundamental reciprocal duality of $n$-dimensional projective geometry which associates to points polar hyperplanes, and more generally $k$-dimensional convex bodies to polar $(n-1-k)$-dimensional convex bodies. It is…
The purpose of this book is to build up the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for the researches of arithmetic geometry in several directions.
A polarity of a projective plane is a map, often assumed to be involutive, mapping a generic point to a generic line and reciprocally. The most classical polarity is the polarity with respect to a conic, but other exist: the harmonic…
Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to…
One of the basic questions in number theory is to determine semi-simple l-adic representations of the absolute Galois group of a number field. In this paper, we discuss the question for two dimensional representations over a totally real…
In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…
Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This…
We introduce a notion of probabilistic convexity and generalize some classical globalization theorems in Alexandrov geometry. A weighted Alexandrov's lemma is developed as a basic tool.
In this paper we describe trigonometry on the de Sitter surface. For that a characterization of geodesics is given, leading to various types of triangles. We define lengths and angles of these. Then, transferring the concept of polar…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
We develop a global cohomology theory for number fields by offering topological cohomology groups, an arithmetical duality, a Riemann-Roch type theorem, and two types of vanishing theorem. As applications, we study moduli spaces of…
Extending the work of Freese, we further develop the theory of generalized trigonometric functions. In particular, we study to what extent the notion of polar form for the complex numbers may be generalized to arbitrary associative…
Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground fields to arbitrary non-archimedean valued fields. To achieve this, we develop…
Deleting a hyperplane from a polar space associated with a symplectic polarity we get a specific, symplectic, affine polar space. Similar geometry, called an \afsempol\ arises as a result of generalization of the notion of an alternating…
Foundational cases of the generalized Stokes' theorem are visualized using geometric algebra. From considering bivector valued fields, two seldom used instances of the theorem are obtained. Graphical representations are given, showing a…
In this article we study multisymplectic geometry, i.e., the geometry of manifolds with a non-degenerate, closed differential form. First we describe the transition from Lagrangian to Hamiltonian classical field theories, and then we…
We study geometric structures arising from Hermitian forms on linear spaces over real algebras beyond the division ones. Our focus is on the dual numbers, the split-complex numbers, and the split-quaternions. The corresponding geometric…