Posts In Category modular
the buckyball curve
on July 2, 2008 by lieven in geometry, groups, modular, Comments (5)
We are after the geometric trinity corresponding to the trinity of exceptional Galois groups
The surfaces on the right have the corresponding group on the left as their group of automorphisms. But, there is a lot more group-theoretic info hidden in the geometry. Before we sketch the case, let us recall the simpler situation [...]
Dedekind or Klein ?
on April 22, 2008 by lieven in groups, modular, numbers, Comments (3)
The black&white psychedelic picture on the left of a tessellation of the hyperbolic upper-halfplane, was called the Dedekind tessellation in this post, following the reference given by John Stillwell in his excellent paper Modular Miracles, The American Mathematical Monthly, 108 (2001) 70-76.
But is this correct terminology? Nobody else uses it apparently. So, let’s [...]
the secret revealed…
on March 26, 2008 by lieven in groups, modular, numbers, Comments (3)
Often, one can appreciate the answer to a problem only after having spend some time trying to solve it, and having failed … pathetically.
When someone with a track-record of coming up with surprising mathematical tidbits like John McKay sends me a mystery message claiming to contain “The secret of Monstrous Moonshine and the universe”, I’m [...]
Monstrous Easter Egg Race
on March 23, 2008 by lieven in groups, modular, Comments (1)
Here’s a sweet Easter egg for you to crack : a mysterious message from none other than the discoverer of Monstrous Moonshine himself…
From: mckayj@Math.Princeton.EDU Date: Mon 10 Mar 2008 07:51:16 GMT+01:00 To: lieven.lebruyn@ua.ac.be
The secret of Monstrous Moonshine and the universe.
Let j(q) = 1/q + 744 + sum( c[k]*q^k,k>=1) be the [...]
the McKay-Thompson series
on March 22, 2008 by lieven in groups, modular, Comments (0)
Monstrous moonshine was born (sometime in 1978) the moment John McKay realized that the linear term in the j-function
is surprisingly close to the dimension of the smallest non-trivial irreducible representation of the monster group, which is 196883. Note that at that time, the Monster hasn’t been constructed yet, and, the only traces of its possible [...]
Farey symbols of sporadic groups
on March 20, 2008 by lieven in geometry, groups, modular, Comments (0)
John Conway once wrote :
There are almost as many different constructions of 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 and starting from (half of) the Farey sequences and the associated cuboid tree diagram [...]
Quiver-superpotentials
on January 14, 2008 by lieven in geometry, modular, Comments (1)
It’s been a while, so let’s include a recap : a (transitive) permutation representation of the modular group is determined by the conjugacy class of a cofinite subgroup , or equivalently, to a dessin d’enfant. We have introduced a quiver (aka an oriented graph) which comes from a triangulation of the compactification of [...]
quivers versus quilts
on January 2, 2008 by lieven in groups, modular, Comments (1)
We have associated to a subgroup of the modular group a quiver (that is, an oriented graph). For example, one verifies that the fundamental domain of the subgroup (an index 3 subgroup) is depicted on the right by the region between the thick lines with the identification of edges as indicated. The [...]
mathematics for 2008 (and beyond)
on December 30, 2007 by lieven in general, modular, Comments (5)
Via the n-category cafe (and just now also the Arcadian functor ) I learned that Benjamin Mann of DARPA has constructed a list of 23 challenges for mathematics for this century.
DARPA is the “Defense Advanced Research Projects Agency” and is an agency of the United States Department of Defense ‘responsible for the development [...]
the modular group and superpotentials (2)
on December 28, 2007 by lieven in geometry, groups, modular, Comments (2)
Last time we have that that one can represent (the conjugacy class of) a finite index subgroup of the modular group by a Farey symbol or by a dessin or by its fundamental domain. Today we will associate a quiver to it.
For example, the modular group itself is represented by the Farey symbol [...]
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