# Posts Tagged: simples

• absolute, stories

## Art and the absolute point

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In his paper Cyclotomy and analytic geometry over $\mathbb{F}_1$ Yuri I. Manin sketches and compares four approaches to the definition of a geometry over $\mathbb{F}_1$, the elusive field with one element. He writes : “Preparing a colloquium talk in Paris, I have succumbed to the temptation to associate them with some dominant trends in the… Read more »

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## Anabelian vs. Noncommutative Geometry

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This is how my attention was drawn to what I have since termed anabelian algebraic geometry, whose starting point was exactly a study (limited for the moment to characteristic zero) of the action of absolute Galois groups (particularly the groups $Gal(\overline{K}/K)$, where K is an extension of finite type of the prime field) on… Read more »

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## Segal’s formal neighbourhood result

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

• stories

## problema bovinum

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Suppose for a moment that some librarian at the Bodleian Library announces that (s)he discovered an old encrypted book attributed to Isaac Newton. After a few months of failed attempts, the code is finally cracked and turns out to use a Public Key system based on the product of two gigantic prime numbers, $2^{32582657}-1$… Read more »

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## M-geometry (3)

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For any finite dimensional A-representation S we defined before a character $\chi(S)$ which is an linear functional on the noncommutative functions $\mathfrak{g}_A = A/[A,A]_{vect}$ and defined via $\chi_a(S) = Tr(a | S)$ for all $a \in A$ We would like to have enough such characters to separate simples, that is we… Read more »

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## M-geometry (1)

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Take an affine $\mathbb{C}$-algebra A (not necessarily commutative). We will assign to it a strange object called the tangent-quiver $\vec{t}~A$, compute it in a few examples and later show how it connects with existing theory and how it can be used. This series of posts can be seen as the promised notes of… Read more »

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## Hexagonal Moonshine (3)

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Hexagons keep on popping up in the representation theory of the modular group and its close associates. We have seen before that singularities in 2-dimensional representation varieties of the three string braid group $B_3$ are ‘clanned together’ in hexagons and last time Ive mentioned (in passing) that the representation theory of the modular group… Read more »

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## Hexagonal Moonshine (2)

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Delving into finite dimensional representations of the modular group $\Gamma = PSL_2(\mathbb{Z})$ it is perhaps not too surprising to discover numerical connections with modular functions. Here, one such strange observation and a possible explanation. Using the _fact_ that the modular group $\Gamma = PSL_2(\mathbb{Z})$ is the free group product $C_2 \ast C_3$… Read more »

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## Hexagonal Moonshine (1)

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Over at the Arcadian Functor, Kea is continuing her series of blog posts on M-theory (the M is supposed to mean either Monad or Motif). A recurrent motif in them is the hexagon and now I notice hexagons popping up everywhere. I will explain some of these observations here in detail, hoping that someone, more… 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 »

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## anabelian geometry

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Last time we saw that a curve defined over $\overline{\mathbb{Q}}$ gives rise to a permutation representation of $PSL_2(\mathbb{Z})$ or one of its subgroups $\Gamma_0(2)$ (of index 2) or $\Gamma(2)$ (of index 6). As the corresponding monodromy group is finite, this representation factors through a normal subgroup of finite index, so it… Read more »

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## noncommutative curves and their maniflds

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Last time we have seen that the noncommutative manifold of a Riemann surface can be viewed as that Riemann surface together with a loop in each point. The extra loop-structure tells us that all finite dimensional representations of the coordinate ring can be found by separating over points and those living at just one point… Read more »