Birational maps as morphisms of cognitive structures

Birational maps and their inverses are defined in terms of rational functions. They are very special in the sense that they map algebraic numbers in a given extension E of rationals to E itself. In the TGD framework, E defines a unique discretization of the space-time surface if the preferred coordinates of the allowed points belong to E. I refer to this discretization as cognitive representation. Birational maps map points in E to points in E so that they define what might be called cognitive morphism.

M⁸-H duality duality (H=M⁴× CP₂) relates the number vision of TGD to the geometric vision. M⁸-H duality maps the 4-surfaces in M⁸_c to space-time surfaces in H: a natural condition is that in some sense it maps E to E and cognitive representations to cognitive representations. There are special surfaces in M⁸_c that allow cognitive explosion in the number-theotically preferred coordinates. M⁴ and hyperbolic spaces H³ (mass shells), which contain 3-surfaces defining holographic data, are examples of these surfaces. Also the 3-D light-like partonic orbits defining holographic data. Possibly also string world sheets define holographic data. Does cognitive explosion happen also in these cases?

In M⁸_c octonionic structure allows to identify natural preferred coordinates. In H, in particular M⁴, the preferred coordinates are not so unique but should be related by birational mappings. So called Hamilton-Jacobi structures define candidates for preferred coordinates: could different Hamilton-Jacobi structures relate to the each other by birational maps? In this article these questions are discussed.

See the article Birational maps as morphisms of cognitive structures or the chapter New findings related to the number theoretical view of TGD.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.