Related papers: A note on Gurzadyan theorem
Without invoking a perturbative expansion, we define the cosmological curvature perturbation, and consider its behaviour assuming that the universe is smooth over a sufficiently large comoving scale. The equations are simple, resembling…
This is a series of lecture notes explaining topos theory and its application in physics.
The past few years have seen dramatic breakthroughs and spectacular and puzzling discoveries in astrophysics and cosmology. In many cases, the new observations can only be explained with the introduction of new fundamental physics. Here we…
Although there are many simple proofs of Jordan's decomposition theorem in the literature (see [1], the references mentioned there, and [2]), our proof seems to be even more elementary. In fact, all we need is the theorem on the dimensions…
We illustrate the concept of mathematical proof.
Dark matter has been introduced to explain many independent gravitational effects at different astronomical scales, in galaxies, groups of galaxies, clusters, superclusters and even across the full horizon. This review describes the…
A review of the principles of observational testing of cosmological theories is given with a special emphasis on the distinction between observational facts and theoretical hypotheses. A classification of modern cosmological theories and…
We analyze the necessary conditions for a body to remain rigid in an expanding cosmological Universe. First, we establish the main theorems and definitions for having a rigid body in a general spacetime as well as the new concept of…
The unsatisfactory status of the search for a consistent and predictive quantization of gravity is taken as motivation to study the question whether geometrical laws could be more fundamental than quantization procedures. In such an…
In this note we give a detailed proof of a theorem of Aubin.
The comparison of the Standard Cosmological Model (SCM) with astronomical observations, i.e. theory versus experiment, and with the Minimal Standard Model (MSM) in particle physics, i.e. theory versus theory, is discussed. The main issue of…
Cosmography is the tool that makes possible to untie the interpretation of cosmological observations from the definition of any dynamical prior. We review the constraints on the cosmographic parameter obtained using the most thorough data…
This short note is intended to review the foundations of mechanics, trying to present them with the greatest mathematical and conceptual clarity. It was attempted to remove most of inessential, even parasitic issues, which can hide the true…
These notes are an exposition of Galois Theory from the original Lagrangian and Galoisian point of view. A particular effort was made here to better understand the connection between Lagrange's purely combinatorial approach and Galois…
The aim of this note is to provide an intrinsic proof of the Gauss--Bonnet theorem without invoking triangulations, which is achieved by exploiting complex structures.
This paper aims to provide a careful and self-contained introduction to the theory of topological degree in Euclidean spaces. It is intended for people mostly interested in analysis and, in general, a heavy background in algebraic or…
There are several mathematical and physical reasons why Dirac's quantization must hold. How far one can go without it remains an open problem. The present work outlines a few steps in this direction.
In cosmology, we would like to explain our observations and predict future observations from theories of the entire universe. Such cosmological theories make ontological assumptions of what entities exist and what their properties and…
The goal of this paper is to experiment new math concepts and theories, especially if they run counter to the classical ones. To prove that contradiction is not a catastrophe, and to learn to handle it in an (un)usual way. To transform the…
This text consists on a series of introductory lectures on cosmology for mathematicians and physicists who are not specialized on the subject.