Related papers: Fun Problems in Geometry and Beyond
This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
Mathematical educational soft explore, investigating in a dynamical way, some algebraically, geometrically problems, the expected results being used to involve a lot of mathematical results. One such software soft is GeoGebra. The software…
We present a variation of the water puzzle, which is related to a simple model of marginal utility. The problem has an intriguing solution and can be extended in several directions.
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also…
We introduce the notions of mixed resolutions and simplicial sections, and prove a theorem relating them. This result is used (in another paper) to study deformation quantization in algebraic geometry.
Thirty research questions on meromorphic functions and complex differential equations are listed and discussed. The main purpose of this paper is to make this collection of problems available to everyone.
The purpose of this article is to motivate the study of invariant, and especially conformally invariant, differential pairings. Since a general theory is lacking, this work merely presents some interesting examples of these pairings,…
The purpose of this paper is to introduce an algebraic cohomology and formal deformation theory of left alternative algebras. Connections to some other algebraic structures are given also.
A pedagogical but concise overview of fiber bundles and their connections is provided, in the context of gauge theories in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing…
During the process of teaching the concept of derivative, it is common and natural to refer to geometric interpretations, such as the use of the tangent line and the maximum and minimum points of a function, to illustrate the scope of the…
Basic concepts and definitions in differential geometry and topology which are important in the theory of solitons and instantons are reviewed. Many examples from soliton theory are discussed briefly, in order to highlight the application…
Choose a polynomial in three variables with not more than three or four monomials of moderate degree. Take simple coefficients as 1 and -1. Then draw a picture of the solution variety in real three space using a ray-tracing program like…
In this paper we collect the main properties of free curves in the complex projective plane and a lot of conjectures and open problems, both old and new. In the quest to understand the mystery of free curves, many tools were developed and…
Diffeology extends differential geometry to spaces beyond smooth manifolds. This paper explores diffeology's key features and illustrates its utility with examples including singular and quotient spaces, and applications in symplectic…
It is shown how to extend the formal variational calculus in order to incorporate integrals of divergences into it. Such a generalization permits to study nontrivial boundary problems in field theory on the base of canonical formalism.
Conformal invariance plays a significant role in many areas of Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an important role in geometry: theory of Riemannian…
In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the finite field analog of questions originally raised in the…
Some scientists take themselves and their work very seriously. However, there are plenty of cases of humour being combined with science. Here I review some examples from the broad fields of physics and astronomy, particularly focusing on…
A differential geometric approach to singular perturbation theory is presented. It is shown that singular perturbation problems such as multiple-scale and boundary layer problems can be treated more easily on a differential geometric basis.…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…