higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
In anabelian geometry one studies how much information about a space $X$ (specifically: an algebraic variety) is contained already in its first étale homotopy group $\pi^{et}_1(X,x)$ (specifically: the algebraic fundamental group).
The term “anabelian” is supposed to be alluding to the fact that “the less abelian $\pi^{et}_1(X,x)$ is, the more information it carries about $X$.” Precisely: an anabelian group is a non-trivial group for which every finite index subgroup has trivial center.
Accordingly, an algebraic variety whose isomorphism class is entirely determined by $\pi^{et}_1(X,x)$ is called an anabelian variety.
An early conjecture motivating the theory (in Grothendieck 84) was that all hyperbolic curves over number fields are anabelian varieties. This was eventually proven by various authors in various cases. In (Uchida) and (Neukirch) it was shown that an isomorphism between Galois groups of number fields implies the existence of an isomorphism between those number fields. For algebraic curves over finite fields, over number fields and over p-adic field the statement was eventually completed by (Mochizuki 96).
Grothendieck also conjectured the existence of higher-dimensional anabelian varieties, but these are still very mysterious.
algebraic fundamental group also called the ‘geometric fundamental group’ by Grothendieck.
child's drawing/ Dessins d’enfant.
The notion of anabelian geometry originates in
written in response to Faltings’ work on the Mordell Conjecture.
There is some discussion of the area in
Surveys include
Yuri Tschinkel, Introduction to anabelian geometry, talk at Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends, Oxford 2014 (slides pdf)
Leila Schneps, page 60 (2) of Grothendieck’s “Long march through Galois theory” (pdf)
A comprehensive introduction is in
Examples are discussed in
The classification of anabelian varieties for number fields was shown in
J. Neukirch, Über die absoluten Galoisgruppen algebraischer Zahlkörper, Journées Arithmétiques de Caen (Univ. Caen, Caen, 1976), pp. 67–79. Asterisque, No. 41-42, Soc. Math. France, Paris (1977)
K. Uchida. Isomorphisms of Galois groups, J. Math. Soc. Japan 28 (1976), no. 4, 617–620.
K. Uchida, Isomorphisms of Galois groups of algebraic function fields, Ann. Math. (2) 106 (1977), no. 3, p. 589–598.
and for algebraic curves in
See also
Frans Oort, Lecture notes. Informal notes (not for publication) made available for the Lorentz Center workshop ‘Anabelian number theory and geometry’, December 3-5, 2001
N. V. Durov, Топологические реализации алгебраических многообразий (Topological realizations of algebraic varieties), preprint POMI 13/2012 (in Russian) abstract, pdf.gz