Related papers: A blueprinted view on $\mathbb F_1$-geometry
Ordered blueprints are algebraic objects that generalize monoids and ordered semirings, and $\mathbb{F}_1^{\pm}$-algebras are ordered blueprints that have an element $\epsilon$ that acts as $-1$. In this work we introduce an analogue of the…
We review the following subjects: 1. Basic theory on algebraic curves and their moduli space, 2. Schottky uniformization theory of Riemann surfaces, and its extension called arithmetic uniformization theory, 3. Application to these theories…
Cyclotomy, the study of cyclotomic classes and cyclotomic numbers, is an area of number theory first studied by Gauss. It has natural applications in discrete mathematics and information theory. Despite this long history, there are…
This work has the purpose of applying the concept of Geometric Calculus (Clifford Algebras) to the Fibre Bundle description of Quantum Mechanics. Thus, it is intended to generalize that formulation to curved spacetimes [the base space of…
This introductory text arises from a lecture given in G\"oteborg, Sweden, given by the first author and is intended for undergraduate students, as well as for any mathematically inclined reader wishing to explore a synthesis of ideas…
Quantum geometry, characterized by the quantum geometric tensor, is pivotal in diverse physical phenomena in quantum materials. In condensed matter systems, quantum geometry refers to the geoemtric properties of Bloch states in the…
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…
The underlying mathematical structures of gauge theories are known to be geometrical in nature and the local and global features of this geometry have been studied for a long time in mathematics under the name of fibre bundles. It is now…
This paper introduces the historical development of the symmetries for describing magnetic structures culminating in the derivation of the black and white and colored space groups. Beginning from the Langevin model of the Curie law, it aims…
A part of Grothendieck's program for studying the Galois group $G_{\mathbb Q}$ of the field of all algebraic numbers $\overline{\mathbb Q}$ emerged from his insight that one should lift its action upon $\overline{\mathbb Q}$ to the action…
The geometry of Quantum Mechanics in the context of uncertainty and complementarity, and probability is explored. We extend the discussion of geometry of uncertainty relations in wider perspective. Also, we discuss the geometry of…
We provide exposition into the field of projection theory, which lies at the intersection of incidence geometry and geometric measure theory. We first give the necessary preliminaries in Chapter 2, focusing on incidences between points and…
This discussion paper presents some parts of the work in progress. It is shown that G.W. Leibniz was the first who raised the question about geometric interpretation of fractional-order operators. Geometric interpretations of the…
Theory of representations of F-algebra is a natural development of the theory of F-algebra. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation. In the book I considered the…
The book can be divided in three parts: the Lagrange geometry of order $k$, presented in the first three chapters, the geometrical theory of the dual manifolds $T^{*k}M$ - chapters 4-7 and the geometry of Hamilton spaces of order $k$ and…
This is an expository treatise on the development of the classical geometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the…
The parameterisation of rotations in three dimensional Euclidean space is an area of applied mathematics that has long been studied, dating back to the original works of Euler in the 18th century. As such, many ways of parameterising a…
k-Contact geometry is a generalisation of contact geometry to analyse field theories. We develop an approach to k-contact geometry based on distributions that are distributionally maximally non-integrable and admit, locally, k commuting…
In this paper, we give a geometrization and a generalization of a lemma of differential Galois theory. This geometrization, in addition of giving a nice insight on this result, offers us the occasion to investigate several points of…
In this paper we introduce congruence spaces, which are topological spaces that are canonically attached to monoid schemes and that reflect closed topological properties. This leads to satisfactory topological characterizations of closed…