Tag: anabelian

  • Mumford’s treasure map

    In the series “Brave new geometries” we give an introduction to ‘strange’ but exciting new ideas. We start with Grothendieck’s scheme-revolution, go on with Soule’s geometry over the field with one element, Mazur’s arithmetic topology, Grothendieck’s anabelian geometry, Connes’ noncommutative geometry etc.

  • This week at F_un Mathematics (1)

    Some links to posts on Soule’s algebraic geometry over the field with one element.

  • recycled : dessins

    In a couple of days I’ll be blogging for 4 years… and I’m in the process of resurrecting about 300 posts from a database-dump made in june. For example here’s my first post ever which is rather naive. This conversion program may last for a couple of weeks and I apologize for all unwanted pingbacks […]

  • Anabelian & Noncommutative Geometry 2

    Last time (possibly with help from the survival guide) we have seen that the universal map from the modular group $\Gamma = PSL_2(\mathbb{Z}) $ to its profinite completion $\hat{\Gamma} = \underset{\leftarrow}{lim}~PSL_2(\mathbb{Z})/N $ (limit over all finite index normal subgroups $N $) gives an embedding of the sets of (continuous) simple finite dimensional representations $\mathbf{simp}_c~\hat{\Gamma} \subset…

  • Anabelian vs. Noncommutative Geometry

    This is how my attention was drawn to what I have since termed anabelian algebraic geometry, whose starting point was exactly a study (limited for the moment to characteristic zero) of the action of absolute Galois groups (particularly the groups $Gal(\overline{K}/K) $, where K is an extension of finite type of the prime field) on…

  • neverendingbooks-geometry

    Here a list of saved pdf-files of previous NeverEndingBooks-posts on geometry in reverse chronological order.

  • anabelian geometry

    Last time we saw that a curve defined over $\overline{\mathbb{Q}} $ gives rise to a permutation representation of $PSL_2(\mathbb{Z}) $ or one of its subgroups $\Gamma_0(2) $ (of index 2) or $\Gamma(2) $ (of index 6). As the corresponding monodromy group is finite, this representation factors through a normal subgroup of finite index, so it…

  • permutation representations of monodromy groups

    Today we will explain how curves defined over $\overline{\mathbb{Q}} $ determine permutation representations of the carthographic groups. We have seen that any smooth projective curve $C $ (a Riemann surface) defined over the algebraic closure $\overline{\mathbb{Q}} $ of the rationals, defines a _Belyi map_ $\xymatrix{C \ar[rr]^{\pi} & & \mathbb{P}^1} $ which is only ramified over…

  • noncommutative bookmarks

    At last, some excitement about noncommutative geometry in the blogosphere. From what I deduce from reading the first posts, Arup Pal set up a new blog called Noncommutative Geometry and subsequently handed it over to Masoud Khalkhali who then got Alain Connes to post on it who, in turn, is asking people to submit posts,…