An elementary survey of mathematical cosmology is presented. We cover certain key ideas and developments in a qualitative way, from the time of the Einstein static universe in 1917 until today. We divide our presentation into four main periods, the first one containing important cosmologies discovered until 1960. The second period (1960–80) contains discussions of geometric extensions of the standard cosmology, singularities, chaotic behaviour and the initial input of particle physics ideas into cosmology. Our survey for the third period (1980–2000) continues with brief descriptions of the main ideas of inflation, the multiverse, quantum, Kaluza-Klein and string cosmologies, wormholes and baby universes, cosmological stability and modified gravity. The last period that ends today includes various more advanced topics such as M-theoretic cosmology, braneworlds, the landscape, topological issues, the measure problem, genericity, dynamical singularities and dark energy. We emphasize certain threads that run throughout the whole period of development of theoretical cosmology and underline their importance in the overall structure of the field. This is Part A of our survey covering the first two periods of development of the subject. The second part will include the third and fourth periods. We end this outline with an inclusion of the abstracts of all papers contributed to the Philosophical Transactions of the Royal Society A theme issue, ‘The Future of Mathematical Cosmology’.
This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.
A set of basic vectors locally describing metric properties of an arbitrary 2-dimensional (2D) surface is used for construction of fundamental algebraic objects having nilpotent and idempotent properties. It is shown that all possible linear combinations of the objects when multiplied behave as a set of hypercomples (in particular, quaternion) units; thus interior structure of the 3D space dimensions pointed by the vector units is exposed. Geometric representations of elementary surfaces (2D-sells) structuring the dimensions are studied in detail. Established mathematical link between a vector quaternion triad treated as a frame in 3D space and elementary 2D-sells prompts to raise an idea of "world screen" having 1/2 of a space dimension but adequately reflecting kinematical properties of an ensemble of 3D frames.
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