# Segal’s formal neighbourhood result

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 a general ringtheoretic gem. After… Read more →

# problema bovinum

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, $2^{32582657}-1$ and $2^{30402457}-1$, which were… Read more →

# M-geometry (3)

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 would like to have an… Read more →

# M-geometry (1)

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 my talks at the GAMAP-workshop… Read more →

# Hexagonal Moonshine (3)

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 is controlled by the double… Read more →

# Hexagonal Moonshine (2)

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 $C_2 \ast C_3$ it is fairly easy to… Read more →

# Hexagonal Moonshine (1)

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 in tune with recent technology,… Read more →

# recap and outlook

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 G of its group of… Read more →

# anabelian geometry

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 makes sense to look at… Read more →

# noncommutative curves and their maniflds

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 are classified by the isoclasses… Read more →

# the noncommutative manifold of a Riemann surface

The natural habitat of this lesson is a bit further down the course, but it was called into existence by a comment/question by Kea I don’t yet quite see where the nc manifolds are, but I guess that’s coming. As I’m enjoying telling about all sorts of sources of finite dimensional representations of $SL_2(\mathbb{Z})$ (and will carry on doing… Read more →

# coalgebras and non-geometry 3

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 there are as many directed… Read more →

# coalgebras and non-geometry 2

Last time we have seen that the _coalgebra of distributions_ of an affine smooth variety is the direct sum (over all points) of the dual to the etale local algebras which are all of the form $\mathbb{C}[[ x_1,\ldots,x_d ]]$ where $d$ is the dimension of the variety. Generalizing this to _non-commutative_ manifolds, the first questions are : “What… Read more →

# non-geometry

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 →

# noncommutative topology (4)

For a qurve (aka formally smooth algebra) A a *block* is a (possibly infinite dimensional over the basefield) left A-module X such that its endomorphism algebra $D = End_A(X)$ is a division algebra and X (considered as a right D-module) is finite dimensional over D. If a block X is finite dimensional over the basefield, we call it a *brick*… Read more →