Tag: non-commutative

  • Penrose tilings and noncommutative geometry

    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, […]

  • changes (ahead)

    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…

  • Langlands versus Connes

    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…

  • Views of noncommutative spaces

    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…

  • F_un with Manin

    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’).…

  • the Manin-Marcolli cave

    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…

  • 2006 paper nominees

    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…

  • coalgebras and non-geometry 2

    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…

  • non-(commutative) geometry

    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…

  • non-geometry

    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…