Posts Tagged: Galois

  • 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 »

  • noncommutative, number theory

    Langlands versus Connes

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    This is a belated response to a Math-Overflow exchange between Thomas Riepe and Chandan Singh Dalawat asking for a possible connection between Connes’ noncommutative geometry approach to the Riemann hypothesis and the Langlands program. Here’s the punchline : a large chunk of the Connes-Marcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as… Read more »

  • games, number theory

    Seating the first few thousand Knights

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    The Knight-seating problems asks for a consistent placing of n-th Knight at an odd root of unity, compatible with the two different realizations of the algebraic closure of the field with two elements.

  • absolute

    big Witt vectors for everyone (1/2)

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    Next time you visit your math-library, please have a look whether these books are still on the shelves : Michiel Hazewinkel‘s Formal groups and applications, William Fulton’s and Serge Lange’s Riemann-Roch algebra and Donald Knutson’s lambda-rings and the representation theory of the symmetric group. I wouldn’t be surprised if one or more of these books… Read more »

  • featured

    The odd knights of the round table

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    Here’s a tiny problem illustrating our limited knowledge of finite fields : “Imagine an infinite queue of Knights ${ K_1,K_2,K_3,\ldots } $, waiting to be seated at the unit-circular table. The master of ceremony (that is, you) must give Knights $K_a $ and $K_b $ a place at an odd root of unity, say $\omega_a… Read more »

  • games, number theory

    On2 : transfinite number hacking

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    Surely Georg Cantor’s transfinite ordinal numbers do not have a real-life importance? Well, think again.

  • web

    best of 2008 (1) : wiskundemeisjes

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    A feeble attempt to translate the Marcolli-post by the ‘wiskundemeisjes’.

  • 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

    Manin’s geometric axis

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    Manin proposes the idea of projecting spec(Z[x]) not only onto spec(Z), but also to a geometric axis by considering the integers as an algebra over the field with one element.

  • geometry, number theory, stories

    Andre Weil on the Riemann hypothesis

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    Some quotes of Andre Weil on the Riemann hypothesis.

  • 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.

  • geometry, groups, number theory

    the buckyball curve

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    We are after the geometric trinity corresponding to the trinity of exceptional Galois groups The surfaces on the right have the corresponding group on the left as their group of automorphisms. But, there is a lot more group-theoretic info hidden in the geometry. Before we sketch the $L_2(11) $ case, let us recall the simpler… Read more »

  • geometry, groups, representations

    Klein’s dessins d’enfant and the buckyball

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    We saw that the icosahedron can be constructed from the alternating group $A_5 $ by considering the elements of a conjugacy class of order 5 elements as the vertices and edges between two vertices if their product is still in the conjugacy class. This description is so nice that one would like to have a… Read more »

  • geometry, groups

    the buckyball symmetries

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    The buckyball is without doubt the hottest mahematical object at the moment (at least in Europe). Recall that the buckyball (middle) is a mixed form of two Platonic solids the Icosahedron on the left and the Dodecahedron on the right. For those of you who don’t know anything about football, it is that other ball-game,… Read more »

  • geometry, groups, math, number theory

    Arnold’s trinities

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    Referring to the triple of exceptional Galois groups $L_2(5),L_2(7),L_2(11) $ and its connection to the Platonic solids I wrote : “It sure seems that surprises often come in triples…”. Briefly I considered replacing triples by trinities, but then, I didnt want to sound too mystic… David Corfield of the n-category cafe and a dialogue on… Read more »

  • groups

    Galois’ last letter

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    “Ne pleure pas, Alfred ! J’ai besoin de tout mon courage pour mourir à vingt ans!” We all remember the last words of Evariste Galois to his brother Alfred. Lesser known are the mathematical results contained in his last letter, written to his friend Auguste Chevalier, on the eve of his fatal duel. Here the… Read more »

  • featured

    adeles and ideles

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    Before we can even attempt to describe the adelic description of the Bost-Connes Hecke algebra and its symmetries, we’d probably better recall the construction and properties of adeles and ideles. Let’s start with the p-adic numbers $\hat{\mathbb{Z}}_p $ and its field of fractions $\hat{\mathbb{Q}}_p $. For p a prime number we can look at the… Read more »

  • featured

    the Bost-Connes Hecke algebra

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    As before, $\Gamma $ is the subgroup of the rational linear group $GL_2(\mathbb{Q}) $ consisting of the matrices $\begin{bmatrix} 1 & b \\ 0 & a \end{bmatrix} $ with $a \in \mathbb{Q}_+ $ and $\Gamma_0 $ the subgroup of all matrices $\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} $ with $n \in \mathbb{N}… Read more »

  • featured

    the crypto lattice

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    Last time we have seen that tori are dual (via their group of characters) to lattices with a Galois action. In particular, the Weil descent torus $R_n=R^1_{\mathbb{F}_{p^n}/\mathbb{F}_p} \mathbb{G}_m $ corresponds to the permutation lattices $R_n^* = \mathbb{Z}[x]/(x^n-1) $. The action of the generator $\sigma $ (the Frobenius) of the Galois group $Gal(\mathbb{F}_{p^n}/\mathbb{F}_p) $ acts on… Read more »

  • featured

    Weil descent

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    A classic Andre Weil-tale is his narrow escape from being shot as a Russian spy The war was a disaster for Weil who was a conscientious objector and so wished to avoid military service. He fled to Finland, to visit Rolf Nevanlinna, as soon as war was declared. This was an attempt to avoid being… Read more »

  • featured

    ECSTR aka XTR

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    The one thing that makes it hard for an outsider to get through a crypto-paper is their shared passion for using nonsensical abbreviations. ECSTR stands for “Efficient Compact Subgroup Trace Representation” and we are fortunate that Arjen Lenstra and Eric Verheul shortened it in their paper The XTR public key system to just XTR. As… Read more »

  • featured

    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 »

  • featured

    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 »

  • stories

    Vacation reading

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    Im in the process of writing/revising/extending the course notes for next year and will therefore pack more math-books than normal. These are for a 3rd year Bachelor course on Algebraic Geometry and a 1st year Master course on Algebraic and Differential Geometry. The bachelor course was based this year partly on Miles Reid’s Undergraduate Algebraic… Read more »

  • featured

    neverendingbooks-geometry (2)

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    Here pdf-files of older NeverEndingBooks-posts on geometry. For more recent posts go here.