Highly structured ring spectrum and the mathematical foundations of E-infinity theory

14th December, 2010.

E-infinity communication No. 60

Highly structured ring spectrum and the mathematical foundations of E-infinity theory

Mohamed El Naschie did not develop his theory starting from abstract mathematics. He could not even if he had wanted to because, as stressed by him on countless occasions, he is not a mathematician and does not hold a degree in mathematics nor in fact in physics. This is not usual but it is not that unusual either. For instance Prof. E. Witten who is a Field Medalist has studied neither mathematics nor physics as an under graduate. The great inventor of green functions was not even an academic and as everyone knows, the best study is self study. All the same a mathematical foundation for E-infinity was given by El Naschie many years ago after the initial geometrical picture of a Cantorian space was established. After realizing that E-infinity theory is a particularly simple and neat example of K-theory and Rene Thom (bordism) spectra El Naschie opted for the name E-infinity suggesting that all these subjects, including infinity-categories are united in a mathematical theory called “Highly structured ring spectrum” which deals with multiplicative processes similar to E-infinity and is usually designated in the mathematical literature by E-infinity rings and E-infinity loop algebra. Prof. El Naschie says that he is particularly indebted to Prof. Sir R. Penrose because his tiling was an eye opener for him. He is equally indebted to the work of Prof. A. Connes because it made him realize the large arsenal of mathematics behind noncommutative geometry and K-theory of which he actually never dreamed when he started working on his simple model of Cantorian-fractal spacetime.
A fairly mathematical paper dealing with the foundations of E-infinity as related to the Coxeter and reflection groups is the following note: ‘Mathematical foundations of E-infinity via Coxeter and reflection groups’, M.S. El Naschie, CS&F, Vol. 37 (2008), pp. 1267-1268. After becoming familiar with the dimensional theory of F. Hausdorff as distinct from that of K. Menger and P. Urysohn, El Naschie started his own mathematical program to introduce transfiniteness everywhere. In doing so he was far ahead of almost all mathematicians because he was of course using intuitive arguments without the essential mathematical rigor which mathematicians must subject themselves to. El Naschie followed two generalizations from integer to non-integer and even irrational values, namely the Hausdorff dimension and the factorial function. The Hausdorff dimension generalizes the usual topological integer dimension to a non-integer ‘fractal’ dimension. Similarly the gamma function generalizes the factorial function to non-integer function. El Naschie did not stop at that and generalized the integer dimension of a symmetry group to a non-integer dimension of a fractal symmetry semi-group. If you think about physics very deeply, like a mathematician of the caliber of A. Connes, then you will find that except in physics integer values are mostly empty mathematical concepts not applicable to the rest of the real world. The quantum Hall effect was discovered by Nobel Laureate K. von Klitzing and generalized to a fractional Hall effect by Nobel Laureate H. Stoermer. El Naschie’s theory predicts a fractal Hall effect as the final generalization of the previous two and he has written a few papers about that and all that remains is the experimental confirmation El Naschie’s fractal Hall effect.
Another mathematical generalization which was achieved within E-infinity theory is the curvature of a fractal-Cantorian point set. This is discussed in connection with the Gauss-Bonnet theorem in one of the most important papers of E-infinity: “On zero-dimensional points curvature in the dynamics of Cantorial-fractal spacetime setting and high energy particle physics”, M.S. El Naschie, CS&F, Vol. 41 (2009), pp. 2725-2732. The detailed discussion of this paper will be given in a forth coming communication.
E—infinity Group.