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A nice interview with Jacques Roubaud (the guy responsible for Bourbaki’s death announcement) in the courtyard of the ENS. He talks about go, categories, the composition of his book $\in$ and, of course, Grothendieck and Bourbaki. Clearly there are pop-math books like dedicated to $\pi$ or $e$, but I don’t know just one novel having… Read more »
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In the spring of 2009 I did spend a fortnight dog-sitting in a huge house in the countryside, belonging to my parents-in-law, who both passed away the year before. That particular day it was raining and thundering heavily. To distract myself from the sombre and spooky atmosphere in the house I began to surf the… Read more »
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Sometimes a MathOverflow question gets deleted before I can post a reply… Yesterday (New-Year) PD1&2 were visiting, so I merely bookmarked the What is the knot associated to a prime?-topic, promising myself to reply to it this morning, only to find out that the page no longer exists. From what I recall, the OP interpreted… Read more »
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Here’s a tiny problem illustrating our limited knowledge of finite fields : “Imagine an infinite queue of Knights ${ K_1,K_2,K_3,\ldots } $, waiting to be seated at the unit-circular table. The master of ceremony (that is, you) must give Knights $K_a $ and $K_b $ a place at an odd root of unity, say $\omega_a… Read more »
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I really like Matilde Marcolli’s idea to use some of Jackson Pollock’s paintings as metaphors for noncommutative spaces. In her talk she used this painting and refered to it (as did I in my post) as : Jackson Pollock “Untitled N.3”. Before someone writes a post ‘The Pollock noncommutative space hoax’ (similar to my own… Read more »
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The general public expects pictures from geometers, even from non-commutative geometers. Hence, it is important for researchers in this topic to make an attempt to convey the mental picture they have of their favourite noncommutative space, … somehow. Two examples : This picture was created by Shahn Majid. It appears on his visions of noncommutative… Read more »
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Conway’s nim-arithmetic on ordinal numbers leads to many surprising identities, for example who would have thought that the third power of the first infinite ordinal equals 2…
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In the series “Brave new geometries” we give an introduction to ‘strange’ but exciting new ideas. We start with Grothendieck’s scheme-revolution, go on with Soule’s geometry over the field with one element, Mazur’s arithmetic topology, Grothendieck’s anabelian geometry, Connes’ noncommutative geometry etc.
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A quick recap of last time. We are trying to make sense of affine varieties over the elusive field with one element $\mathbb{F}_1 $, which by Grothendieck’s scheme-philosophy should determine a functor $\mathbf{nano}(N)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto N(A) $ from finite Abelian groups to sets, typically giving pretty small sets $N(A) $. Using the… Read more »
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Last time we have seen how an affine $\mathbb{C} $-algebra R gives us a maxi-functor (because the associated sets are typically huge) $\mathbf{maxi}(R)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto Hom_{\mathbb{C}-alg}(R, \mathbb{C} A) $ Substantially smaller sets are produced from finitely generated $\mathbb{Z} $-algebras S (therefore called mini-functors) $\mathbf{mini}(S)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto Hom_{\mathbb{Z}-alg}(S, \mathbb{Z} A)… Read more »
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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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Monstrous moonshine was born (sometime in 1978) the moment John McKay realized that the linear term in the j-function $j(q) = \frac{1}{q} + 744 + 196884 q + 21493760 q^2 + 864229970 q^3 + \ldots $ is surprisingly close to the dimension of the smallest non-trivial irreducible representation of the monster group, which is 196883…. Read more »
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John Conway once wrote : There are almost as many different constructions of $M_{24} $ as there have been mathematicians interested in that most remarkable of all finite groups. In the inguanodon post Ive added yet another construction of the Mathieu groups $M_{12} $ and $M_{24} $ starting from (half of) the Farey sequences and… Read more »
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Juan de Mairena linked in the comments to last post to a truly great retro-chess problem ! In the position below white is to play and mate in three! At first this seems wrong as there is an obvious mate in two : 1. Qe2-f1, Kh1xh2 2. Rg3-h3 The ingenious point being that black claims… Read more »
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Ever tried a chess problem like : White to move, mate in two! Of course you have, and these are pretty easy to solve : you only have to work through the finite list of white first moves and decide whether or not black has a move left preventing mate on the next white move…. Read more »
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MUBs (for Mutually Unbiased Bases) are quite popular at the moment. Kea is running a mini-series Mutual Unbias as is Carl Brannen. Further, the Perimeter Institute has a good website for its seminars where they offer streaming video (I like their MacromediaFlash format giving video and slides/blackboard shots simultaneously, in distinct windows) including a talk… Read more »
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Time to wrap up this series on the Bost-Connes algebra. Here’s what we have learned so far : the convolution product on double cosets $\begin{bmatrix} 1 & \mathbb{Z} \\ 0 & 1 \end{bmatrix} \backslash \begin{bmatrix} 1 & \mathbb{Q} \\ 0 & \mathbb{Q}_{> 0} \end{bmatrix} / \begin{bmatrix} 1 & \mathbb{Z} \\ 0 & 1 \end{bmatrix} $… Read more »
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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 »
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This semester, I’m running a 3rd year course on Marcus du Sautoy’s The music of the primes. The concept being that students may suggest topics, merely sketched in the book, and then we’ll go a little deeper into them. I’ve been rather critical about the book before, but, rereading it last week (and knowing a… Read more »
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Ibrahim Belkadi, one of my first-year group theory students invented the micro-sudokube, that is, a cube having a solution to a micro-sudoku on all its sides such that these solutions share one row along an edge. For example, here are all the solutions for a given central solution. There are 4 of them with ${… Read more »
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The adelic interpretation of the Bost-Connes Hecke algebra $\mathcal{H} $ is based on three facts we’ve learned so far : The diagonal embedding of the rational numbers $\delta~:~\mathbb{Q} \rightarrow \prod_p \mathbb{Q}_p $ has its image in the adele ring $\mathcal{A} $. ( details ) There is an exact sequence of semigroups $1 \rightarrow \mathcal{G} \rightarrow… Read more »
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Oystein Ore mentions the following puzzle from Brahma-Sphuta-Siddhanta (Brahma’s Correct System) by Brahmagupta : An old woman goes to market and a horse steps on her basket and crashes the eggs. The rider offers to pay for the damages and asks her how many eggs she had brought. She does not remember the exact number,… Read more »
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Before we can even attempt to describe the adelic description of the Bost-Connes Hecke algebra and its symmetries, we’d probably better recall the construction and properties of adeles and ideles. Let’s start with the p-adic numbers $\hat{\mathbb{Z}}_p $ and its field of fractions $\hat{\mathbb{Q}}_p $. For p a prime number we can look at the… Read more »
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Towards the end of the Bost-Connes for ringtheorists post I freaked-out because I realized that the commutation morphisms with the $X_n^* $ were given by non-unital algebra maps. I failed to notice the obvious, that algebras such as $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] $ have plenty of idempotents and that this mysterious ‘non-unital’ morphism was nothing else but multiplication… Read more »
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Over the last days I’ve been staring at the Bost-Connes algebra to find a ringtheoretic way into it. Ive had some chats about it with the resident graded-guru but all we came up with so far is that it seems to be an extension of Fred’s definition of a ‘crystalline’ graded algebra. Knowing that several… Read more »
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