PlanetPhysics/Anabelian Geometry

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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. The latter two areas 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 and physical mathematics developments may expand specific Anabelian Geometry applications to quantum geometry and Quantum Algebraic Topology.
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Anabelian Geometry

The area of mathematics called Anabelian Geometry (AAG) began with Alexander Grothendieck's introduction of the term in his seminal and influential report "Esquisse d'un Programme" (1) produced in 1980. The basic setting of his anabelian geometry is that of the algebraic fundamental group ${\displaystyle {\mathcal {G}}}$  of an algebraic variety ${\displaystyle X}$  (which is a basic concept in Algebraic Geometry), and also possibly a more generally defined, but related, geometric object. The algebraic fundamental group , ${\displaystyle {\mathcal {G}}}$ , in this case determines how the algebraic variety ${\displaystyle X}$  can be mapped into, or linked to, another geometric object ${\displaystyle Y}$ , assuming that ${\displaystyle {\mathcal {G}}}$  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 \htmladdnormallink{groupoids {http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} and fundamental groupoids} of a topological space (that generalize the concept of fundamental group), as well as from that of \htmladdnormallink{higher dimensional algebra {http://planetphysics.us/encyclopedia/HigherDimensionalAlgebra2.html} (HDA)}. Thus, the fundamental anabelian question posed by Grothendieck was, and is: "how much information about the isomorphism class of the variety ${\displaystyle X}$  is contained in the knowledge of the etale fundamental group?" (on p. 2 in ${\displaystyle http://www.math.jussieu.fr/~leila/SchnepsLM.pdf}$ ). At this point, stepping down from the general, abstract setting of the Anabelian Geometry it would be useful to consider a specific, concrete example.

A Concrete Example

In the case of curves, ${\displaystyle C}$  , these could be either affine (as in Einstein's or Weyl's approaches to General Relativity), or projective , as in a variety ${\displaystyle V}$ . Consider here a specific hyperbolic curve ${\displaystyle H}$  , that is defined as the complement of ${\displaystyle n}$  points in a projective algebriac curve of genus ${\displaystyle g}$  , which is assumed to be both smooth and irreducible, and also defined over a field ${\displaystyle K}$  (that is finitely generated over its prime field ) such that: ${\displaystyle 2-2g-n<0}$ . Grothendieck conjectured in 1979 that the fundamental group ${\displaystyle {\mathcal {G}}}$  of ${\displaystyle C}$  , which is a profinite group, determines the curve ${\displaystyle C}$  itself, or that the isomorphism class of ${\displaystyle {\mathcal {G}}}$  determines the isomorphism class of ${\displaystyle C}$  ; this also points towards a conjecture regarding the natural equivalence ${\displaystyle \eta }$  of their corresponding categories.

Generalizations

Much more elaborate, generalizations of Grothendieck's Anabelian Geometry are posible by considering higher-dimensional, ${\displaystyle pro-l}$ , ${\displaystyle Hom}$ -- versions, and so on, involving for example fundamental groupoids and fundamental double groupoids (2).

References

1. Alexander Grothendieck. 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.3--48; English transl., ibid., pp. 243--283.

2. Jochen Koenigsmann. 2001. Anabelian geometry over almost arbitrary fields. ${\displaystyle http://www.uni-math.gwdg.de/tschinkel/SS04/koenigsmann2.pdf}$

3. S. Mochizuki, H. Nakamura, A. Tamagawa. "The Grothendieck Conjecture on the fundamental groups of algebraic curves", Sugaku Expositions , {\mathbf 14}(1), (2001), 31--53.

{\mathbf...work in progress}