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\begin{document}
This is a new topic in which the \htmladdnormallink{Anabelian Geometry}{http://planetphysics.us/encyclopedia/IsomorphismClass.html} approach will be defined and compared with other appoaches that are disticnt from it such as
\htmladdnormallink{non-Abelian algebraic topology}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} and \emph{\htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}}{ http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}. The latter two \htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} have already made an impact on \htmladdnormallink{quantum theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} 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 \htmladdnormallink{general relativity}{http://planetphysics.us/encyclopedia/SR.html} on \htmladdnormallink{spacetimes}{http://planetphysics.us/encyclopedia/SR.html} that are more general than pseudo- or quasi- Riemannian `spaces'. Furthermore, other \htmladdnormallink{theoretical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} developments may expand specific Anabelian Geometry applications to \htmladdnormallink{quantum geometry}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html} and \htmladdnormallink{Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html}.
\section{Anabelian Geometry}
The area of mathematics called {\em Anabelian Geometry (AAG)} began with Alexander Grothendieck's introduction of the term in his seminal and influential \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} {\em ``Esquisse d'un Programme''} $[1]$ produced in 1980. The basic setting of his anabelian geometry is that of the \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{fundamental group}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html} $\mathcal{G}$ of an \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} variety $X$ (which is a basic \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} in Algebraic Geometry), and also possibly a more generally defined, but related, geometric \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}. The {\em algebraic fundamental group}, $\mathcal{G}$, in this case determines how the \htmladdnormallink{algebraic variety}{http://planetphysics.us/encyclopedia/IsomorphismClass.html} $X$ can be mapped into, or linked to, another geometric \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $Y$, assuming that $\mathcal{G}$ is {\em \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}} or \htmladdnormallink{noncommutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}. This specific approach differs significantly, of course, from that of Noncommutative Geometry introduced by Alain Connes. It also differs from the main-stream \htmladdnormallink{nonabelian algebraic topology}{http://planetphysics.us/encyclopedia/NAQAT2.html} (NAAT)'s generalized approach to topology in terms of \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} and \htmladdnormallink{fundamental groupoids}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} of a \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} space (that generalize the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of fundamental space), as well as from that of \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} (\htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html}). Thus, the fundamental \htmladdnormallink{anabelian}{http://planetphysics.us/encyclopedia/IsomorphismClass.html} question posed by Grothendieck was, and is:
{\em ``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 $$http://www.math.jussieu.fr/~leila/SchnepsLM.pdf$$ ).
At this point, stepping down from the general, abstract setting of the Anabelian Geometry it would be useful to consider a specific, concrete example.
\subsection{A Concrete Example}
In the case of curves, {\em $C$}, these could be either {\em affine} (as in \htmladdnormallink{Einstein's}{http://planetphysics.us/encyclopedia/AlbertEinstein.html} or Weyl's approaches to General Relativity), or {\em projective}, as in a variety $V$. Consider here a specific hyperbolic curve {\em $H$}, that is defined as the complement of $n$ points in a {\em projective algebriac curve of genus $g$}, which is assumed to be both smooth and irreducible, and also defined over a \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} $K$ (that is finitely generated over its {\em prime field}) such that: $$2-2g-n < 0$$. Grothendieck conjectured in 1979 that the \htmladdnormallink{fundamental group}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html} $\mathcal{G}$ of {\em $C$}, which is a {\em profinite \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}}, determines the curve {\em $C$} itself, or that the {\em \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} class of $\mathcal{G}$ determines the \htmladdnormallink{isomorphism class}{http://planetphysics.us/encyclopedia/IsomorphismClass.html} of {\em $C$}; this also points towards a conjecture regarding the {\em \htmladdnormallink{natural equivalence}{http://planetphysics.us/encyclopedia/IsomorphismClass.html}} $\eta$ of their corresponding \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html}.
\subsection{Generalizations}
Much more elaborate, generalizations of Grothendieck's Anabelian Geometry are posible by considering higher-dimensional, $pro-l$, $Hom$-- versions, and so on, involving for example \htmladdnormallink{fundamental groupoids}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} and fundamental \htmladdnormallink{double groupoids}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} $[2]$.
\subsection{References}
1. \htmladdnormallink{Alexander Grothendieck}{http://planetphysics.us/encyclopedia/AlexanderGrothendieck.html}, 1984. "Esquisse d'un Programme", (1984 manuscript), published in "Geometric Galois Actions", L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997, pp. 5--48; English transl., ibid., pp. 243--283
2. Jochen Koenigsmann.2001. Anabelian geometry over almost arbitrary fields
$http://www.uni-math.gwdg.de/tschinkel/SS04/koenigsmann2.pdf$
3. S. Mochizuki, H. Nakamura, A. Tamagawa. ``The Grothendieck Conjecture
on the fundamental groups of algebraic curves'', {\em Sugaku Expositions}
{\bf 14}(1), (2001), 31--53.
{\bf\htmladdnormallink{...work}{http://planetphysics.us/encyclopedia/Work.html} in progress}
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