Related papers: Mathematics, physics and philosophy in Riemann's w…
The works of R. Descartes, I. M. Gelfand and A. Grothendieck have convinced us that commutative rings should be thought of as rings of functions on some appropriate (commutative) spaces. If we try to push this notion forward we reach the…
We present a pedagogical introduction to the recent advances in the computational geometry, physical implications, and data science of Calabi-Yau manifolds. Aimed at the beginning research student and using Calabi-Yau spaces as an exciting…
This paper is a commentary and a reading guide to three papers by Herbert Busemann, \"Uber die Geometrien, in denen die "Kreise mit unendlichem Radius" die k\"urzesten Linien sind." (On the geometries where circles of infinite radius are…
Richard P. Feynman's work on gravitation, as can be inferred from several published and unpublished sources, is reviewed. Feynman was involved with this subject at least from late 1954 to the late 1960s, giving several pivotal contributions…
Modified gravity theories have received increased attention lately due to combined motivation coming from high-energy physics, cosmology and astrophysics. Among numerous alternatives to Einstein's theory of gravity, theories which include…
The primary aim of this chapter is, commemorating the 150th anniversary of Riemann's death, to explain how the idea of {\it Riemann sum} is linked to other branches of mathematics. The materials I treat are more or less classical and…
This paper is based on a talk presented by the first author at the Short Program on Riemannian Geometry that took place at the Centre de Recherche Math\'ematiques, Universit\'e de Montr\'eal, during the period June 28-July 16, 2004. It is a…
We feel that non-commutative geometry is to particle physics what Riemannian geometry is to gravity. We try to explain this feeling.
This Thesis is devoted to the study of Metric-Affine Theories of Gravity and Applications to Cosmology. The thesis is organized as follows. In the first Chapter we define the various geometrical quantities that characterize a non-Riemannian…
This PhD dissertation is concerned with integral geometric inverse problems. The geodesic ray transform is an operator that encodes the line integrals of a function along geodesics. The dissertation establishes many conditions when such…
Motivated in part by the bi-gravity approach to massive gravity, we introduce and study the multimetric Finsler geometry. For the case of an arbitrary number of dimensions, we study some general properties of the geometry in terms of its…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
The recognition that physical space (or space-time) is curved is a product of the general theory of relativity, such as dramatically shown by the 1919 solar eclipse measurements. However, the mathematical possibility of non-Euclidean…
This essay traces the history of three interconnected strands. Firstly, changes in the concept of number, secondly, the study of the qualities of number, which evolved into number theory, and thirdly, the nature of mathematics itself, from…
This invited essay belongs to a series considering highly influential articles published by the International Journal of Theoretical Physics. In this paper, we highlight the physical content and the profound consequences of Richard…
Group Theory has become an invaluable tool in the physics community. Despite numerous introductory books, the subject remains challenging for beginners. Mathematica has emerged as a popular tool for research and education, offering various…
The results summarized here are intended as rigorous mathematical statements on various physical models coming from condensed matter physics, statistical mechanics (classical and quantum), quantum field theory and cold atoms physics. The…
The aim of this article is to present a comparative review of Riemannian and Finsler geometry. The structures of cut and conjugate loci on Riemannian manifolds have been discussed by many geometers including H. Busemann, M. Berger and W.…
These notes study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. They are based on introductory lectures given at Stony Brook…
The Riemann Hypothesis has been of central interest to mathematicians for a long time and many unsuccessful attempts have been made to either prove or disprove it. Since the Riemann zeta function is defined as a sum of the infinite number…