Related papers: What is Positive Geometry?
The work consists of solutions of metric problems for convex and finite subsets of geodesic spaces.
This survey article is an introduction to Diophantine Geometry at a basic undergraduate level. It focuses on Diophantine Equations and the qualitative description of their solutions rather than detailed proofs.
Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from…
By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…
We discuss an experimental approach to open problems in toric geometry: are smooth projective toric varieties (i) projectively normal and (ii) defined by degree 2 equations? We discuss the creation of lattice polytopes defining smooth toric…
A very short, non-technical review of the main problems which are presently open in the context of the gravitational theory and of its cosmological applications, and which are currently under investigation and deep discussion. We will talk…
We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the…
In this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory. They could give work for future mathematical physicists trained with the…
We first review the historical developments, both in physics and in mathematics, that preceded (and in some sense provided the background of) deformation quantization. Then we describe the birth of the latter theory and its evolution in the…
This is an elementary geometrical proof of Birkhoff theorem. It is hardly important, but the pictures behind are quite nice.
Positive geometries were introduced by Arkani-Hamed--Bai--Lam as a method of computing scattering amplitudes in theoretical physics. We show that a positive geometry from a polytope admits a log resolution of singularities to another…
This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.
The spectral geometry of negatively curved manifolds has received more attention than its positive curvature counterpart. In this paper we will survey a variety of spectral geometry results that are known to hold in the context of…
Traditional geometry employs idealized concepts like that of a point or a curve, the operational definition of which relies on the availability of classical point particles as probes. Real, physical objects are quantum in nature though,…
Mathematical research is often motivated by the desire to reach a beautiful result or to prove it in an elegant way. Mathematician's work is thus strongly influenced by his aesthetic judgments. However, the criteria these judgments are…
In this paper we make some considerations about using a dynamic geometry software for teaching history of mathematica at university level. After a short introduction to the software GeoGebra, we discuss four activities. The first is an…
The problem of constructing a quantum theory of gravity is considered from a novel viewpoint. It is argued that any consistent theory of gravity should incorporate a relational character between the matter constituents of the theory. In…
In this paper I present a comparison theorem for the waist of Riemannian manifolds with positive sectional curvature. The main theorem of this paper gives a partial positive answer to a conjecture formulated by M.Gromov in [8]. The content…
Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…
The problem of advancing coordinatization of mathematics is considered. The need to develop a theory for measuring value and complexity of mathematical implications and proofs is discussed including motivations, benefits and implementation…