# Posts Tagged: Kontsevich

• web

## best of 2008 (2) : big theorems

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

• web

## best of 2008 (1) : wiskundemeisjes

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A feeble attempt to translate the Marcolli-post by the ‘wiskundemeisjes’.

• geometry, noncommutative

## noncommutative F_un geometry (2)

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

• absolute, geometry, stories

## F_un with Manin

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Amidst all LHC-noise, 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 non-existent field with one element $\mathbb{F}_1$ (in French called ‘F-un’)…. Read more »

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## what does the monster see?

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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 »

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## recap and outlook

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After a lengthy spring-break, 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 »

• stories

## noncommutative geometry : a medieval science?

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

• web

## 2006 paper nominees

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

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## coalgebras and non-geometry 3

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Last time we saw that the _coalgebra of distributions_ of a noncommutative manifold can be described as a coalgebra Takeuchi-equivalent 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 »

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## coalgebras and non-geometry

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

• stories

## non-(commutative) geometry

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Now that my non-geometry post is linked via the comments in this string-coffee-table post which in turn is available through a trackback from the Kontsevich-Soibelman paper it is perhaps useful to add a few links. The little I’ve learned from reading about Connes-style non-commutative geometry is this : if you have a situation where a… Read more »

• stories

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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 »

• 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

## a good day at the arxiv

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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 »

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## noncommutative topology (1)

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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 »

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## From Galois to NOG

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Evariste Galois (1811-1832) must rank pretty high on the all-time 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 »

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## a cosmic Galois group

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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 »

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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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## 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 one quiver for GL(2,Z)

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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 quasi-free algebra in the terminology of Cuntz-Quillen or a formally smooth algebra in the terminology of Kontsevich-Rosenberg) a quiver $Q(A)$ and a… 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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## more noncommutative manifolds

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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 »

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## Borcherds’ monster papers

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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 Non-commutative smooth spaces : It seems plausible that Borcherds’ infinite rank algebras with Monstrous symmetry can be realized inside Hall-Ringel algebras for some small smooth… Read more »

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## NOG master class update

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