Conjugacy classes of finite index subgroups of the modular group $\Gamma = PSL_2(\mathbb{Z}) $ are determined by a combinatorial gadget : a modular quilt. By this we mean a finite connected graph drawn on a Riemann surface such that its vertices are either black or white. Moreover, every edge in the graph connects a black to a white vertex and… Read more →

# Tag Archive for representations

# 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 →

# neverendingbooks-geometry (2)

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

# NeverEndingBooks-groups

Here a collection of pdf-files of NeverEndingBooks-posts on groups, in reverse chronological order. Read more →

# down with determinants

The categorical cafe has a guest post by Tom Leinster Linear Algebra Done Right on the book with the same title by Sheldon Axler. I haven’t read the book but glanced through his online paper Down with determinants!. Here is ‘his’ proof of the fact that any n by n matrix A has at least one eigenvector. Take a vector… 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 →

# THE rationality problem

This morning, Esther Beneish arxived the paper The center of the generic algebra of degree p that may contain the most significant advance in my favourite problem for over 15 years! In it she claims to prove that the center of the generic division algebra of degree p is stably rational for all prime values p. Let me begin by… Read more →

# devilish symmetries

In another post we introduced Minkowski’s question-mark function, aka the devil’s straircase and related it to Conways game of _contorted fractions_. Side remark : over at Good Math, Bad Math Mark Chu-Carroll is running a mini-series on numbers&games, so far there is a post on surreal numbers, surreal arithmetic and the connection with games but probably this series will go… 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 →

# permutation representations of monodromy groups

Today we will explain how curves defined over $\overline{\mathbb{Q}} $ determine permutation representations of the carthographic groups. We have seen that any smooth projective curve $C $ (a Riemann surface) defined over the algebraic closure $\overline{\mathbb{Q}} $ of the rationals, defines a _Belyi map_ $\xymatrix{C \ar[rr]^{\pi} & & \mathbb{P}^1} $ which is only ramified over the three points $\\{ 0,1,\infty… 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 →

# The cartographers’ groups

Just as cartographers like Mercator drew maps of the then known world, we draw dessins d ‘enfants to depict the associated algebraic curve defined over $\overline{\mathbb{Q}} $. In order to see that such a dessin d’enfant determines a permutation representation of one of Grothendieck’s cartographic groups, $SL_2(\mathbb{Z}), \Gamma_0(2) $ or $\Gamma(2) $ we need to have realizations of these groups… Read more →

# Monsieur Mathieu

Even a virtual course needs an opening line, so here it is : Take your favourite $SL_2(\mathbb{Z}) $-representation Here is mine : the permutation presentation of the Mathieu group(s). Emile Leonard Mathieu is remembered especially for his discovery (in 1861 and 1873) of five sporadic simple groups named after him, the Mathieu groups $M_{11},M_{12},M_{22},M_{23} $ and $M_{24} $. These were… Read more →

# noncommutative geometry : a medieval science?

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 floor, appears to have involved… Read more →