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From 1899 to 1945,
Elie Cartan, a son of a blacksmith, developed a set of extraordinary
mathematical ideas that have yet to be fully exploited in the physical and
technical sciences. This WWW site is dedicated to certain applications of
Cartan's methods to problems of dissipative, radiative,
irreversible systems. Although
emphasis herein has been placed on hydrodynamic, thermodynamic, and
electromagnetic applications, Cartan's techniques can be used on micro and
cosmological scales as well.
Cartan was the inventor of Spinors, spaces with
Torsion
, a champion of Projective Geometries , and the developer of a system of
calculus called
Exterior Differential Forms.
This remarkable calculus goes beyond the geometrical limitations of Tensor Analysis with its restrictions to diffeomorphisms, for Cartan's exterior calculus has Topological content in both its irreducible (Pfaff dimension) representations, and in its harmonic components (deRham period integrals). Moreover, exterior differential forms are well behaved under functional substitution and the pullbacks with respect to maps that are not even homeomorphic. Therefore differential forms can be used to study topological evolution, where standard tensor methods on contravariant objects fail.
The philosophy to be developed herein is that most visible physical measurements are recognitions of Topological Defects and that irreversibility and biological aging are expressions of Topological Evolution .
For some more history,
click here.
The book by M A Akivis and B Rosenfeld, Elie Cartan (1869-1951) (Providence
R.I., 1993),
and translated by V. Goldberg, is exceptional.