
A comment to Charles Siegel’s ‘big theorems’series got me checking my stats.

A feeble attempt to translate the Marcollipost by the ‘wiskundemeisjes’.

We use Kontsevich’s idea of thin varieties to define complexified varieties over F\_un.

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 »

The Monster is the largest of the 26 sporadic simple groups and has order 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 = 2^46 3^20 5^9 7^6 11^2 13^3 17 19 23 29 31 41 47 59 71. It is not so much the size… Read more »

After a lengthy springbreak, let us continue with our course on noncommutative geometry and $SL_2(\mathbb{Z}) $representations. Last time, we have explained Grothendiecks mantra that all algebraic curves defined over number fields are contained in the profinite compactification $\widehat{SL_2(\mathbb{Z})} = \underset{\leftarrow}{lim}~SL_2(\mathbb{Z})/N $ of the modular group $SL_2(\mathbb{Z}) $ and in the knowledge of a certain subgroup… Read more »

According to a science article in the New York Times, archeologists have discovered “signs of advanced math” in medieval mosaics. An example of a quasicrystalline Penrose pattern was found at the Darbi Imam shrine in Isfahan, Iran. A new study shows that the Islamic patternmaking process, far more intricate than the laying of one‚Äôs bathroom… Read more »

At last, some excitement about noncommutative geometry in the blogosphere. From what I deduce from reading the first posts, Arup Pal set up a new blog called Noncommutative Geometry and subsequently handed it over to Masoud Khalkhali who then got Alain Connes to post on it who, in turn, is asking people to submit posts,… 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 saw that the _coalgebra of distributions_ of a noncommutative manifold can be described as a coalgebra Takeuchiequivalent to the path coalgebra of a huge quiver. This infinite quiver has as its vertices the isomorphism classes of finite dimensional simple representations of the qurve A (the coordinate ring of the noncommutative manifold) and… Read more »

In this series of posts I’ll try to make at least part of the recent [KontsevichSoibelman paper](http://www.arxiv.org/abs/math.RA/0606241) a bit more accessible to algebraists. In nongeometry, the algebras corresponding to *smooth affine varieties* I’ll call **qurves** (note that they are called **quasifree algebras** by Cuntz & Quillen and **formally smooth** by Kontsevich). By definition, a qurve… 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 »

This is not going to be the post I should be writing (this morning I found out that the last post must have been rather cryptic as I didnt manage to get it explained to people who should know at least half of the picture, so at the moment Im writing out a short note… 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 »

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 »

Evariste Galois (18111832) must rank pretty high on the alltime list of moving last words. Galois was mortally wounded in a duel he fought with Perscheux d\’Herbinville on May 30th 1832, the reason for the duel not being clear but certainly linked to a girl called Stephanie, whose name appears several times as a marginal… Read more »

Are there hidden relations between mathematical and physical constants such as $\frac{e^2}{4 \pi \epsilon_0 h c} \sim \frac{1}{137} $ or are these numerical relations mere accidents? A couple of years ago, Pierre Cartier proposed in his paper A mad day’s work : from Grothendieck to Connes and Kontsevich : the evolution of concepts of space… Read more »

OK! I asked to get sidetracked by comments so now that there is one I’d better deal with it at once. So, is there any relation between the noncommutative (algebraic) geometry based on formally smooth algebras and the noncommutative _differential_ geometry advocated by Alain Connes? Short answers to this question might be (a) None whatsoever!… Read more »

Now that the preparation for my undergraduate courses in the first semester is more or less finished, I can begin to think about the courses I’ll give this year in the master class noncommutative geometry. For a change I’d like to introduce the main ideas and concepts by a very concrete example : Ginzburg’s coadjointorbit… Read more »

Before the vacation I finished a rewrite of the One quiver to rule them all note. The main point of that note was to associate to any qurve $A$ (formerly known as a quasifree algebra in the terminology of CuntzQuillen or a formally smooth algebra in the terminology of KontsevichRosenberg) a quiver $Q(A)$ and a… Read more »

Today Travis Schedler posted a nice paper on the arXiv “A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver”. I heard the first time about necklace Lie algebras from Jacques Alev who had heard a talk by Kirillov who constructed an infinite dimensional Lie algebra on the monomials in two noncommuting… Read more »

Can it be that one forgets an entire proof because the result doesn’t seem important or relevant at the time? It seems the only logical explanation for what happened last week. Raf Bocklandt asked me whether a classification was known of all group algebras l G which are noncommutative manifolds (that is, which are formally… Read more »

Yesterday morning I thought that I could use some discussions I had a week before with Markus Reineke to begin to make sense of one sentence in Kontsevich’ Arbeitstagung talk Noncommutative smooth spaces : It seems plausible that Borcherds’ infinite rank algebras with Monstrous symmetry can be realized inside HallRingel algebras for some small smooth… Read more »

Yesterday I made a preliminary program for the first two months of the masterclass noncommutative geometry. It is likely that the program will still undergo changes as at the moment I included only the minicourses given by Bernhard Keller and Markus Reineke but several other people have already agreed to come and give a talk…. Read more »
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