Posts Tagged: topology

  • stories

    the birthday of the primes=knots analogy

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    Last time we discovered that the mental picture to view prime numbers as knots in $S^3$ was first dreamed up by David Mumford. Today, we’ll focus on where and when this happened. 3. When did Mazur write his unpublished preprint? According to his own website, Barry Mazur did write the paper Remarks on the Alexander… Read more »

  • stories

    Who dreamed up the primes=knots analogy?

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    One of the more surprising analogies around is that prime numbers can be viewed as knots in the 3-sphere $S^3$. The motivation behind it is that the (etale) fundamental group of $\pmb{spec}(\mathbb{Z}/(p))$ is equal to (the completion) of the fundamental group of a circle $S^1$ and that the embedding $\pmb{spec}(\mathbb{Z}/(p)) \subset \pmb{spec}(\mathbb{Z})$ embeds this circle… Read more »

  • web

    changes (ahead)

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    In view or recents events & comments, some changes have been made or will be made shortly : categories : Sanitized the plethora of wordpress-categories to which posts belong. At the moment there are just 5 categories : ‘stories’ and ‘web’ (for all posts with low math-content) and three categories ‘level1’, ‘level2’ and ‘level3’, loosely… Read more »

  • 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 »

  • featured

    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 »

  • web

    Lists 2010 : MathOverflow bookmarks

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    A few MathOverflow threads I bookmarked in 2010 for various reasons. Honest answer : Applications of algebraic geometry over a field with one element. James Borger’s answer : “I’m confident that the answer to the original question is no. There are hardly any theorems at all in the subject, much less ones with external applications!… 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 »

  • books, stories

    The artist and the mathematician

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    Over the week-end I read The artist and the mathematician (subtitle : The story of Nicolas Bourbaki, the genius mathematician who never existed) by Amir D. Aczel. Whereas the central character of the book should be Bourbaki, it focusses more on two of Bourbaki’s most colorful members, André Weil and Alexander Grothendieck, and the many… 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

    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.

  • featured

    Mumford’s treasure map

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    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.

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    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 »

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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 »

  • featured

    M-geometry (2)

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    Last time we introduced the tangent quiver $\vec{t}~A $ of an affine algebra A to be a quiver on the isoclasses of simple finite dimensional representations. When $A=\mathbb{C}[X] $ is the coordinate ring of an affine variety, these vertices are just the points of the variety $X $ and this set has the extra structure… Read more »

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    neverendingbooks-geometry (2)

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

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    neverendingbooks-geometry

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

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    the Manin-Marcolli cave

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    Yesterday, Yuri Manin and Matilde Marcolli arXived their paper Modular shadows and the Levy-Mellin infinity-adic transform which is a follow-up of their previous paper Continued fractions, modular symbols, and non-commutative geometry. They motivate the title of the recent paper by : In [MaMar2](http://www.arxiv.org/abs/hep-th/0201036), these and similar results were put in connection with the so called… Read more »

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    the noncommutative manifold of a Riemann surface

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    The natural habitat of this lesson is a bit further down the course, but it was called into existence by a comment/question by Kea I don’t yet quite see where the nc manifolds are, but I guess that’s coming. As I’m enjoying telling about all sorts of sources of finite dimensional representations of $SL_2(\mathbb{Z}) $… Read more »

  • stories

    mathematics & unhappiness

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    Sociologists are a constant source of enlightenment as CNN keeps reminding Kids who are turned off by math often say they don’t enjoy it, they aren’t good at it and they see little point in it. Who knew that could be a formula for success? The nations with the best scores have the least happy,… Read more »

  • stories

    Krull & Paris

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    The Category-Cafe ran an interesting post The history of n-categories claiming that “mathematicians’ histories are largely ‘Royal-road-to-me’ accounts” To my mind a key difference is the historians’ emphasis in their histories that things could have turned out very differently, while the mathematicians tend to tell a story where we learn how the present has emerged… Read more »

  • featured

    noncommutative topology (4)

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    For a qurve (aka formally smooth algebra) A a *block* is a (possibly infinite dimensional over the basefield) left A-module X such that its endomorphism algebra $D = End_A(X)$ is a division algebra and X (considered as a right D-module) is finite dimensional over D. If a block X is finite dimensional over the basefield,… Read more »

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    noncommutative topology (3)

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    For finite dimensional hereditary algebras, one can describe its noncommutative topology (as developed in part 2) explicitly, using results of Markus Reineke in The monoid of families of quiver representations. Consider a concrete example, say $A = \begin{bmatrix} \mathbb{C} & V \\ 0 & \mathbb{C} \end{bmatrix}$ where $V$ is an n-dimensional complex vectorspace, or equivalently,… Read more »

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    a noncommutative topology 2

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    A *qurve* is an affine algebra such that $~\Omega^1~A$ is a projective $~A~$-bimodule. Alternatively, it is an affine algebra allowing lifts of algebra morphisms through nilpotent ideals and as such it is the ‘right’ noncommutative generalization of Grothendieck’s smoothness criterium. Examples of qurves include : semi-simple algebras, coordinate rings of affine smooth curves, hereditary orders… Read more »