Posts Tagged: profinite

  • stories

    the Reddit (after)effect

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    Sunday january 2nd around 18hr NeB-stats went crazy. Referrals clarified that the post ‘What is the knot associated to a prime?’ was picked up at Reddit/math and remained nr.1 for about a day. Now, the dust has settled, so let’s learn from the experience. A Reddit-mention is to a blog what doping is to a… Read more »

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    What is the knot associated to a prime?

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    Sometimes a MathOverflow question gets deleted before I can post a reply… Yesterday (New-Year) PD1&2 were visiting, so I merely bookmarked the What is the knot associated to a prime?-topic, promising myself to reply to it this morning, only to find out that the page no longer exists. From what I recall, the OP interpreted… Read more »

  • absolute, geometry, number theory

    Mazur’s knotty dictionary

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    The algebraic fundamental group of a scheme gives the Mazur-Kapranov-Reznikov dictionary between primes in number fields and knots in 3-manifolds.

  • absolute, geometry, noncommutative

    noncommutative F_un geometry (1)

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    We propose to extend the Connes-Consani definition to noncommuntative F_un varieties.

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    Connes-Consani for undergraduates (3)

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    A quick recap of last time. We are trying to make sense of affine varieties over the elusive field with one element $\mathbb{F}_1 $, which by Grothendieck’s scheme-philosophy should determine a functor $\mathbf{nano}(N)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto N(A) $ from finite Abelian groups to sets, typically giving pretty small sets $N(A) $. Using the… Read more »

  • absolute, groups

    F_un and braid groups

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    Recall that an n-braid consists of n strictly descending elastic strings connecting n inputs at the top (named 1,2,…,n) to n outputs at the bottom (labeled 1,2,…,n) upto isotopy (meaning that we may pull and rearrange the strings in any way possible within 3-dimensional space). We can always change the braid slightly such that we… Read more »

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    Bost-Connes for ringtheorists

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    Over the last days I’ve been staring at the Bost-Connes algebra to find a ringtheoretic way into it. Ive had some chats about it with the resident graded-guru but all we came up with so far is that it seems to be an extension of Fred’s definition of a ‘crystalline’ graded algebra. Knowing that several… Read more »

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    Anabelian & Noncommutative Geometry 2

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    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… Read more »

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    profinite groups survival guide

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    Even if you don’t know the formal definition of a profinte group, you know at least one example which explains the concept : the Galois group of the algebraic numbers $Gal = Gal(\overline{\mathbb{Q}}/\mathbb{Q}) $ aka the absolute Galois group. By definition it is the group of all $\mathbb{Q} $-isomorphisms of the algebraic closure $\overline{\mathbb{Q}} $…. Read more »

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    Anabelian vs. Noncommutative Geometry

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    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… Read more »

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    recap and outlook

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    After a lengthy spring-break, let us continue with our course on noncommutative geometry and $SL_2(\mathbb{Z}) $-representations. Last time, we have explained Grothendiecks mantra that all algebraic curves defined over number fields are contained in the profinite compactification $\widehat{SL_2(\mathbb{Z})} = \underset{\leftarrow}{lim}~SL_2(\mathbb{Z})/N $ of the modular group $SL_2(\mathbb{Z}) $ and in the knowledge of a certain subgroup… Read more »

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    anabelian geometry

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    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… Read more »