Tag Archive for differential


Arnold’s trinities

Referring to the triple of exceptional Galois groups $L_2(5),L_2(7),L_2(11) $ and its connection to the Platonic solids I wrote : “It sure seems that surprises often come in triples…”. Briefly I considered replacing triples by trinities, but then, I didnt want to sound too mystic… David Corfield of the n-category cafe and a dialogue on infinity (and perhaps other blogs… Read more →


It’s been a while, so let’s include a recap : a (transitive) permutation representation of the modular group $\Gamma = PSL_2(\mathbb{Z}) $ is determined by the conjugacy class of a cofinite subgroup $\Lambda \subset \Gamma $, or equivalently, to a dessin d’enfant. We have introduced a quiver (aka an oriented graph) which comes from a triangulation of the compactification of… Read more →

Vacation reading

Im in the process of writing/revising/extending the course notes for next year and will therefore pack more math-books than normal. These are for a 3rd year Bachelor course on Algebraic Geometry and a 1st year Master course on Algebraic and Differential Geometry. The bachelor course was based this year partly on Miles Reid’s Undergraduate Algebraic Geometry and partly on David… Read more →

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 “holography” principle in modern theoretical… Read more →

noncommutative bookmarks

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, such as todays post by… Read more →


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 years ago in Antwerp. There… Read more →


Thanks to Andrei Sobolevskii for his comment pointing me to a wonderful initiative : CiteULike. What is CiteULike? CiteULike is a free service to help academics to share, store, and organise the academic papers they are reading. When you see a paper on the web that interests you, you can click one button and have it added to your personal… Read more →


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 make a CarbonCopyClooner- image of… 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 non-commutative differential forms and symplectic forms) so I\’d better update the Granada notes soon as they will also be… 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 $\tilde{Q}$ $\xymatrix{\vtx{} \ar@/^/[rr]^{a=u+du} \ar@/_/[rr]_{b=u-du} &… Read more →

differential forms

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 vertex-idempotents, see the post on [path algebras][2]… Read more →

cotangent bundles

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 && \vtx{} \ar@(u,ur)^x \ar@(d,dr)_{x^*} \ar@/^/[ll]^{a^*}}… Read more →

moduli spaces

In [the previous part][1] we saw that moduli spaces of suitable representations of the quiver $\xymatrix{\vtx{} \ar[rr] & & \vtx{} \ar@(ur,dr)} $ locally determine the moduli spaces of vectorbundles over smooth projective curves. There is yet another classical problem related to this quiver (which also illustrates the idea of looking at families of moduli spaces rather than individual ones) :… Read more →

algebraic vs. differential nog

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