PlanetPhysics/Nutest

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This is a new topic in which the Anabelian Geometry approach will be defined and compared with other appoaches that are disticnt from it such as

non-Abelian algebraic topology and noncommutative geometry{ http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}. The latter two fields have already made an impact on quantum theories that seek a new setting beyond SUSY--the Standard Model of modern physics. Moreover, it is also possible to consider in this topic novel, possible approaches to relativity theories, especially to general relativity on spacetimes that are more general than pseudo- or quasi- Riemannian `spaces'. Furthermore, other theoretical physics developments may expand specific Anabelian Geometry applications to quantum geometry and Quantum Algebraic Topology.

Anabelian Geometry

The area of mathematics called Anabelian Geometry (AAG) began with Alexander Grothendieck's introduction of the term in his seminal and influential work "Esquisse d'un Programme" $[1]$ produced in 1980. The basic setting of his anabelian geometry is that of the algebraic fundamental group ${\mathcal {G}}$ of an algebraic variety $X$ (which is a basic concept in Algebraic Geometry), and also possibly a more generally defined, but related, geometric object. The algebraic fundamental group , ${\mathcal {G}}$, in this case determines how the algebraic variety $X$ can be mapped into, or linked to, another geometric object $Y$, assuming that ${\mathcal {G}}$ is non-Abelian or noncommutative. This specific approach differs significantly, of course, from that of Noncommutative Geometry introduced by Alain Connes. It also differs from the main-stream nonabelian algebraic topology (NAAT)'s generalized approach to topology in terms of groupoids and fundamental groupoids of a topological space (that generalize the concept of fundamental space), as well as from that of higher dimensional algebra (HDA). Thus, the fundamental anabelian question posed by Grothendieck was, and is: "how much information about the isomorphism class of the variety $X$ is contained in the knowledge of the etale fundamental group?" (on p. 2 in $http://www.math.jussieu.fr/~leila/SchnepsLM.pdf$ ).