Related papers: Geometrization in Geometry
In the last two decades, one of the most important developments in Riemannian geometry is the collapsing theory of Cheeger-Fukaya-Gromov. A Riemannian manifold is called (sufficiently) collapsed if its dimension looks smaller than its…
We reformulate superalgebra and supergeometry in completely categorical terms by a consequent use of the functor of points. The increased abstraction of this approach is rewarded by a number of great advantages. First, we show that one can…
In this article we present a generalization of a Leibniz's geometrical theorem and an application of it.
This paper attempts to relate some ideas of Grothendieck in his Esquisse d'un programme and some of the recent results on 2-dimensional topology and geometry. Especially, we shall discuss the Teichm\"uller theory, the mapping class groups,…
Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place. We show an elementary algebraic approach to…
This is a gentle introduction to a general theory of universal polynomials associated to classification of map-germs, called Thom polynomials. The theory was originated by Ren\'e Thom in the 1950s and has since been evolved in various…
This paper reexamines univariate reduction from a toric geometric point of view. We begin by constructing a binomial variant of the $u$-resultant and then retailor the generalized characteristic polynomial to fully exploit sparsity in the…
Not any geometry can be axiomatized. The paradoxical Godel's theorem starts from the supposition that any geometry can be axiomatized and goes to the result, that not any geometry can be axiomatized. One considers example of two close…
Mathematical proof is undoubtedly the cornerstone of mathematics. The emergence, in the last years, of computing and reasoning tools, in particular automated geometry theorem provers, has enriched our experience with mathematics immensely.…
This paper gives an overview of several key innovations in the 19th century which led to complex geometry in the 20th century. This includes the creation of the complex plane, the work of Abel on addition theorems for generalized elliptic…
Local-to-global principles are spread all-around in mathematics. The classical Cartan-Hadamard Theorem from Riemannian geometry was generalized by W. Ballmann for metric spaces with non-positive curvature, and by S. Alexander and R. Bishop…
We formulate a number of new results in Algebraic Geometry and outline their derivation from Theorem 2.12 which belongs to Algebraic Combinatorics.
Given a complex structure, we investigate diverging sequences of projective structures on the fixed complex structure in terms of Thurston's parametrization. In particular, we will give a geometric proof to the theorem by Kapovich stating…
Grafting is a surgery on Riemann surfaces introduced by Thurston which connects hyperbolic geometry and the theory of projective structures on surfaces. We will discuss the space of projective structures in terms of the Thurston's geometric…
We survey the development of Clifford's geometric algebra and some of its engineering applications during the last 15 years. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly…
Perhaps the most significant, if not the most important, achievements in chemistry and physics are the Periodic Table of the Elements in Chemistry and the Standard Model of Elementary Particles in Physics. A comparable achievement in…
It is well known that the geometrical framework of Riemannian geometry that underlies general relativity and its torsionful extension to Riemann-Cartan geometry can be obtained from a procedure known as gauging the Poincare algebra.…
In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to…
The geometry conjecture, which was posed nearly a quarter of a century ago, states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain…
This paper provides an overview of modern digital geometry and topology through mathematical principles, algorithms, and measurements. It also covers recent developments in the applications of digital geometry and topology including image…