
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 »

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

This is a belated response to a MathOverflow 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 ConnesMarcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as… Read more »

The general public expects pictures from geometers, even from noncommutative 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 »

Amidst all LHCnoise, 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 nonexistent field with one element $\mathbb{F}_1 $ (in French called ‘Fun’)…. Read more »

Yesterday, Yuri Manin and Matilde Marcolli arXived their paper Modular shadows and the LevyMellin infinityadic transform which is a followup of their previous paper Continued fractions, modular symbols, and noncommutative geometry. They motivate the title of the recent paper by : In [MaMar2](http://www.arxiv.org/abs/hepth/0201036), these and similar results were put in connection with the so called… Read more »

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

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 _noncommutative_ manifolds, the… Read more »

Now that my nongeometry post is linked via the comments in this stringcoffeetable post which in turn is available through a trackback from the KontsevichSoibelman paper it is perhaps useful to add a few links. The little I’ve learned from reading about Connesstyle noncommutative geometry is this : if you have a situation where a… Read more »

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

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 »

Let us take a hopeless problem, motivate why something like noncommutative 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 YangBaxter equation) All such solutions (for varying size of matrices) form an additive Abelian category… Read more »

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 »

The third volume in the NeverEndingBooksseries 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 NeverEndingBooksURL, I… Read more »

Here the story of an idea to construct new examples of noncommutative 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 quasifree algebras, aka formally smooth algebras) are the \’affine\’ pieces of noncommutative… Read more »

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 »

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 noncommutative algebra. Here is the programme of that section Session 1 (Friday 20 May) — chair : Jacques Alev (Univ. Reims) 15.3016.25… Read more »

Unlike the cooler people out there, I haven’t received my _preordered_ 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 »

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 HTMLpages. But, eventually I… Read more »

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 noncommutative differential forms and symplectic forms) so I\’d better update the Granada notes soon… Read more »

Here is the construction of this normal space or chart . The subsemigroup 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 nonisomorphic onedimensional solutions in In particular, in any component containing an open subset of representations corresponding to… Read more »

Now, can we assign such an noncommutative 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 eigenspaces. For example where with g the generator of . Similarly, where is a primitive… Read more »

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 noncommutative compactification… Read more »

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 noncommutative) $C$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 »

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 »
Close