Posts Tagged: non-commutative

  • noncommutative

    Penrose tilings and noncommutative geometry

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    Penrose tilings are aperiodic tilings of the plane, made from 2 sort of tiles : kites and darts. It is well known (see for example the standard textbook tilings and patterns section 10.5) that one can describe a Penrose tiling around a given point in the plane as an infinite sequence of 0’s and 1’s,… 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 »

  • 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 »

  • featured

    Views of noncommutative spaces

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    The general public expects pictures from geometers, even from non-commutative geometers. Hence, it is important for researchers in this topic to make an attempt to convey the mental picture they have of their favourite noncommutative space, … somehow. Two examples : This picture was created by Shahn Majid. It appears on his visions of noncommutative… Read more »

  • absolute, geometry, stories

    F_un with Manin

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    Amidst all LHC-noise, Yuri I. Manin arXived today an interesting paper Cyclotomy and analytic geometry over $\mathbb{F}_1 $. The paper gives a nice survey of the existent literature and focusses on the crucial role of roots of unity in the algebraic geometry over the non-existent field with one element $\mathbb{F}_1 $ (in French called ‘F-un’)…. Read more »

  • featured

    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 »

  • web

    2006 paper nominees

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    Here are my nominees for the 2006 paper of the year award in mathematics & mathematical physics : in math.RA : math.RA/0606241 : Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I by Maxim Kontsevich and Yan Soibelman. Here is the abstract : We develop geometric approach to A-infinity algebras and A-infinity categories based… Read more »

  • featured

    coalgebras and non-geometry 2

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    Last time we have seen that the _coalgebra of distributions_ of an affine smooth variety is the direct sum (over all points) of the dual to the etale local algebras which are all of the form $\mathbb{C}[[ x_1,\ldots,x_d ]] $ where $d $ is the dimension of the variety. Generalizing this to _non-commutative_ manifolds, the… Read more »

  • stories

    non-(commutative) geometry

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    Now that my non-geometry post is linked via the comments in this string-coffee-table post which in turn is available through a trackback from the Kontsevich-Soibelman paper it is perhaps useful to add a few links. The little I’ve learned from reading about Connes-style non-commutative geometry is this : if you have a situation where a… Read more »

  • stories

    non-geometry

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    Here’s an appeal to the few people working in Cuntz-Quillen-Kontsevich-whoever noncommutative geometry (the one where smooth affine varieties correspond to quasi-free or formally smooth algebras) : let’s rename our topic and call it non-geometry. I didn’t come up with this term, I heard in from Maxim Kontsevich in a talk he gave a couple of… Read more »

  • web

    a good day at the arxiv

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    The arXiv is a bit like cable tv : on certain days there seems to be nothing interesting on, whereas on others it’s hard to decide what to see in real time and what to record for later. Today was one of the better days, at least on the arXiv. Pavel Etingof submitted the notes… Read more »

  • featured

    why nag? (1)

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    Let us take a hopeless problem, motivate why something like non-commutative algebraic geometry might help to solve it, and verify whether this promise is kept. Suppose we want to know all solutions in invertible matrices to the braid relation (or Yang-Baxter equation) All such solutions (for varying size of matrices) form an additive Abelian category… Read more »

  • featured

    noncommutative topology (1)

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    A couple of days ago Ars Mathematica had a post Cuntz on noncommutative topology pointing to a (new, for me) paper by Joachim Cuntz A couple of years ago, the Notices of the AMS featured an article on noncommutative geometry a la Connes: Quantum Spaces and Their Noncommutative Topology by Joachim Cuntz. The hallmark of… Read more »

  • stories

    work in progress

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    The third volume in the NeverEndingBooks-series will be written by Geert Van de Weyer and will be about (double) Poisson structures in the noncommutative world. Volume 4&5 are becoming clearer every day and if you think you have a project fitting in this series, you can always email to [info@neverenedingbooks.org][3]. As for the NeverEndingBooks-URL, I… Read more »

  • featured

    sexing up curves

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    Here the story of an idea to construct new examples of non-commutative compact manifolds, the computational difficulties one runs into and, when they are solved, the white noise one gets. But, perhaps, someone else can spot a gem among all gibberish… [Qurves](http://www.neverendingbooks.org/toolkit/pdffile.php?pdf=/TheLibrary/papers/qaq.pdf) (aka quasi-free algebras, aka formally smooth algebras) are the \’affine\’ pieces of non-commutative… Read more »

  • stories

    back

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    If you recognize where this picture was taken, you will know that I\’m back from France. If you look closer you will see two bikes, my own Bulls mountainbike in front and Stijn\’s lightweight bike behind. If you see the relative position of the saddles, you will know that Stijn is at least 20 cm… Read more »

  • stories

    hectic days

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    Hectic days ahead! Today, there is the Ph.D. defense of Stijn Symens and the following two days there is a meeting in Ghent where Jacques Alev and me organize a special session on non-commutative algebra. Here is the programme of that section Session 1 (Friday 20 May) — chair : Jacques Alev (Univ. Reims) 15.30-16.25… Read more »

  • stories

    nostalgia

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    Unlike the cooler people out there, I haven’t received my _pre-ordered_ copy (via AppleStore) of Tiger yet. Partly my own fault because I couldn’t resist the temptation to bundle up with a personalized iPod Photo! The good news is that it buys me more time to follow the housecleaning tips. First, my idea was to… Read more »

  • web

    GMD

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    I’m always extremely slow to pick up a trend (let alone a hype), in mathematics as well as in real life. It took me over a year to know of the existence of _blogs_ and to realize that they were a much easier way to maintain a webpage than manually modifying HTML-pages. But, eventually I… Read more »

  • web

    pdfsync

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    I expect to be writing a lot in the coming months. To start, after having given the course once I noticed that I included a lot of new material during the talks (mainly concerning the component coalgebra and some extras on non-commutative differential forms and symplectic forms) so I\’d better update the Granada notes soon… Read more »

  • featured

    why nag? (3)

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    Here is the construction of this normal space or chart . The sub-semigroup of (all dimension vectors of Q) consisting of those vectors satisfying the numerical condition is generated by six dimension vectors, namely those of the 6 non-isomorphic one-dimensional solutions in In particular, in any component containing an open subset of representations corresponding to… Read more »

  • featured

    why nag? (2)

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    Now, can we assign such an non-commutative tangent space, that is a for some quiver Q, to ? As we may restrict any solution in to the finite subgroups and . Now, representations of finite cyclic groups are decomposed into eigen-spaces. For example where with g the generator of . Similarly, where is a primitive… Read more »

  • featured

    seen this quiver?

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    The above quiver on 10 vertices is not symmetric, but has the interesting property that every vertex has three incoming and three outgoing arrows. If you have ever seen this quiver in another context, please drop me a line. My own interest for it is that it is the ‘one quiver’ for a non-commutative compactification… Read more »

  • featured

    B for bricks

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    Last time we argued that a noncommutative variety might be an _aggregate_ which locally is of the form $\mathbf{rep}~A$ for some affine (possibly non-commutative) $C$-algebra $A$. However, we didn't specify what we meant by 'locally' as we didn't define a topology on $\mathbf{rep}~A$, let alone on an arbitrary aggregate. Today we will start the construction… Read more »

  • stories

    writing

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    A long time ago Don Passman told me the simple “secret” for writing books : “Get up and, before you do anything else, try to write 2 or 3 pages. If you do this every day, by the end of the year you’ll have a pretty thick book.” Probably the best advice ever for those… Read more »