Related papers: From Euclid to BGL
We discuss new problems in universal algebraic geometry and explain them by boolean equations.
A survey of new geometric flows motivated by string theories is provided. Their settings can range from complex geometry to almost-complex geometry to symplectic geometry. From the PDE viewpoint, many of them can be viewed as intermediate…
Gravity with incorporation of additional dimensions and noncommutative geometry.
This is a introductory survey of some recent developments of "Galois ideas" in Arithmetic, Complex Analysis, Transcendental Number Theory and Quantum Field Theory, and of some of their interrelations.
In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between non-birational spaces, and realizing geometries in novel fashions. Examples of gauged linear sigma models with…
In recent years, the intersection of algebra, geometry, and combinatorics with particle physics and cosmology has led to significant advances. Central to this progress is the twofold formulation of the study of particle interactions and…
One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…
We introduce an extension of the Standard Model and General Relativity built upon the principle of local conformal invariance, which represents a generalization of a previous work by Bars, Steinhardt and Turok. This is naturally realized by…
Since the end of the 19th century, and after the works of F. Klein and H. Poincar\'e, it is well known that models of elliptic geometry and hyperbolic geometry can be given using projective geometry, and that Euclidean geometry can be seen…
The Euclidean interpretation of special relativity which has been suggested by the author is a formulation of special relativity in ordinary 4D Euclidean space-time geometry. The natural and geometrically intuitive generalization of this…
Historically, there have been many attempts to produce an appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's…
A dynamical non-Abelian 2-form gauge theory (with B \wedge F term) is endowed with the "scalar" and "vector" gauge symmetry transformations. In our present endeavor, we exploit the latter gauge symmetry transformations and perform the…
Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point,…
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…
Differential calculus on the quantum quaternionic group GL(1,H$_q$) is introduced.
The purpose of this thesis is to use the language of orbifold groupoids to describe the geometry and topology of orbifolds, highlighting advantages and disadvantages of this language as they arise.
Symmetry plays an important role in the topological states of quantum matter. Cartan found three symmetry classes, Altland and Zirnbauer extended it to the ten-fold symmetry class for mesoscopic systems. We observe emergence of a new…
This article is a short nontechnical survey of recent progresses in fluid dynamics and differential geometry, relating a conjecture of Lars Onsager to the work of Nash on isometric embeddings.
This paper describes several key discoveries in the 19th century that led to the modern theory of manifolds in the twentieth century: intrinsic differential geometry, projective geometry and higher dimensional manifolds and Riemannian…
Many classical facts in Riemannian geometry have their pseudo-Riemannian analogs. For instance, the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian…