
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 When did Mazur write his unpublished preprint?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 »

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 »

If you ever sat through a lecture by Alain Connes you will know about his insistence on the ‘canonical dynamic nature of noncommutative manifolds’. If you haven’t, he did write a blog post Heart bit 1 about it. I’ll try to explain here that there is a definite “supplément d’âme” obtained in the transition from… Read more »

Yesterday, Ed Segal gave a talk at the Arts. His title “Superpotential algebras from 3fold singularities” didnt look too promising to me. And sure enough it was all there again : stringtheory, Dbranes, CalabiYaus, superpotentials, all the pseudophysics babble that spreads virally among the youngest generation of algebraists and geometers. Fortunately, his talk did contain… Read more »

The problem with criticizing others is that you have to apply the same standards to your own work. So, as of this afternoon, I do agree with all those who said so before : my book is completely unreadable and should either be dumped or entirely rewritten! Here’s what happened : Last week I did… 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 »

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 »

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 »

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 »

[Last time][1] we saw that the algebra $(\Omega_V~C Q,Circ)$ of relative differential forms and equipped with the Fedosov product is again the path algebra of a quiver $\tilde{Q}$ obtained by doubling up the arrows of $Q$. In our basic example the algebra map $C \tilde{Q} \rightarrow \Omega_V~C Q$ is clarified by the following picture of… Read more »

The previous post in this sequence was [(co)tangent bundles][1]. Let $A$ be a $V$algebra where $V = C \times \ldots \times C$ is the subalgebra generated by a complete set of orthogonal idempotents in $A$ (in case $A = C Q$ is a path algebra, $V$ will be the subalgebra generated by the vertexidempotents, see… Read more »

The previous post in this sequence was [moduli spaces][1]. Why did we spend time explaining the connection of the quiver $Q~:~\xymatrix{\vtx{} \ar[rr]^a & & \vtx{} \ar@(ur,dr)^x} $ to moduli spaces of vectorbundles on curves and moduli spaces of linear control systems? At the start I said we would concentrate on its _double quiver_ $\tilde{Q}~:~\xymatrix{\vtx{} \ar@/^/[rr]^a… 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 »

The previous post can be found [here][1]. Pierre Gabriel invented a lot of new notation (see his book [Representations of finite dimensional algebras][2] for a rather extreme case) and is responsible for calling a directed graph a quiver. For example, $\xymatrix{\vtx{} \ar@/^/[rr] & & \vtx{} \ar@(u,ur) \ar@(d,dr) \ar@/^/[ll]} $ is a quiver. Note than it… 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 »

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 »

Bill Schelter was a remarkable man. First, he was a topclass mathematician. If you allow yourself to be impressed, read his proof of the ArtinProcesi theorem. Bill was also among the first to take noncommutative geometry seriously. Together with Mike Artin he investigated a notion of noncommutative integral extensions and he was the first to… Read more »

Yesterday I made reservations for lecture rooms to run the master class on noncommutative geometry sponsored by the ESFNOG project. We have a lecture room on monday and wednesday afternoon and friday the whole day which should be enough. I will run two courses in the program : noncommutative geometry and projects in noncommutative geometry… Read more »
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