Posts Tagged: differential

  • geometry, groups, math, number theory

    Arnold’s trinities

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    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… Read more »

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    Quiver-superpotentials

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    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… Read more »

  • stories

    Vacation reading

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    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… Read more »

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    neverendingbooks-geometry (2)

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    Here pdf-files of older NeverEndingBooks-posts on geometry. For more recent posts go here.

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    the Manin-Marcolli cave

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    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… Read more »

  • web

    noncommutative bookmarks

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    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 »

  • stories

    non-geometry

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    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… Read more »

  • web

    citeUlike

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    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… Read more »

  • stories

    nostalgia

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    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… Read more »

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    pdfsync

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    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… Read more »

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    curvatures

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    [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 »

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    differential forms

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    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… Read more »

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    cotangent bundles

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    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 »

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    moduli spaces

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    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… Read more »

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    algebraic vs. differential nog

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    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!… Read more »

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    path algebras

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    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 »

  • stories

    driven by ambition and sloth

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    Here’s a part of yesterday’s post by bitch ph.d. : But first of all I have to figure out what the hell I’m going to teach my graduate students this semester, and really more to the point, what I am not going to bother to try to cram into this class just because it’s my… Read more »

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    nog course outline

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    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 non-commutative geometry. For a change I’d like to introduce the main ideas and concepts by a very concrete example : Ginzburg’s coadjoint-orbit… Read more »

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    the necklace Lie bialgebra

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    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 non-commuting… Read more »

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    noncommutative geometry

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    Today I did prepare my lectures for tomorrow for the NOG master-class on non-commutative geometry. I\’m still doubting whether it is worth TeXing my handwritten notes. Anyway, here is what I will cover tomorrow : – Examples of l-algebras (btw. l is an arbitrary field) : matrix-algebras, group-algebras lG of finite groups, polynomial algebras, free… Read more »