PlanetPhysics/Nutest

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

[[../ModuleAlgebraic/|non-Abelian algebraic topology]] and [[../NoncommutativeGeometry4/|noncommutative geometry]]{ http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}. The latter two [[../CosmologicalConstant2/|fields]] have already made an impact on [[../SpaceTimeQuantizationInQuantumGravityTheories/|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 [[../SR/|general relativity]] on [[../SR/|spacetimes]] that are more general than pseudo- or quasi- Riemannian `spaces'. Furthermore, other [[../PhysicalMathematics2/|theoretical physics]] developments may expand specific Anabelian Geometry applications to [[../TriangulationMethodsForQuantizedSpacetimes2/|quantum geometry]] and [[../TriangulationMethodsForQuantizedSpacetimes2/|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/|work]] "Esquisse d'un Programme" $[1]$ produced in 1980. The basic setting of his anabelian geometry is that of the [[../CoIntersections/|algebraic]] [[../ModuleAlgebraic/|fundamental group]] $𝒢$ of an [[../CoIntersections/|algebraic]] variety $X$ (which is a basic [[../PreciseIdea/|concept]] in Algebraic Geometry), and also possibly a more generally defined, but related, geometric [[../TrivialGroupoid/|object]]. The algebraic fundamental group , $𝒢$, in this case determines how the [[../IsomorphismClass/|algebraic variety]] $X$ can be mapped into, or linked to, another geometric [[../TrivialGroupoid/|object]] $Y$, assuming that $𝒢$ is [[../AbelianCategory3/|non-Abelian]] or [[../AbelianCategory3/|noncommutative]]. This specific approach differs significantly, of course, from that of Noncommutative Geometry introduced by Alain Connes. It also differs from the main-stream [[../NonabelianAlgebraicTopology3/|nonabelian algebraic topology]] ([[../NonabelianAlgebraicTopology3/|NAAT]])'s generalized approach to topology in terms of [[../GroupoidHomomorphism2/|groupoids]] and [[../CubicalHigherHomotopyGroupoid/|fundamental groupoids]] of a [[../CoIntersections/|topological]] space (that generalize the [[../PreciseIdea/|concept]] of fundamental space), as well as from that of [[../InfinityGroupoid/|higher dimensional algebra]] ([[../InfinityGroupoid/|HDA]]). Thus, the fundamental [[../IsomorphismClass/|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 $http://www.math.jussieu.fr/leila/SchnepsLM.pdf$ ).