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Author: lievenlb

capita selecta

Published April 22, 2004 by lievenlb


Rather than going to the NOG
III Workshop I think it is more fun to give a talk for the Capita
Selecta-course for 2nd year students on “Monstrous Moonshine”. If
I manage to explain to them at least something, I think I am in good
shape for next year\’s Baby Geometry (first year) course. Besides,
afterwards I may decide to give some details of Borcherds\’ solution next year in my 3rd year
Geometry-course…(but this may just be a little bit
over-optimistic).
Anyway, this is what I plan to do in my
lecture : explain both sides of the McKay-observation
that

196 884 = 196 883 + 1

that is, I\’ll give
the action of the modular group on the upper-half plane and prove that
its fundamental domain is just C using the modular j-function (left hand
side) and sketch the importance of the Monster group and its
representation theory (right hand side). Then I\’ll mention Ogg\’s
observation that the only subgroups Gamma(0,p)+ of SL(2,Z)
for which the fundamental domain has genus zero are the prime divisors
p of teh order of the Monster and I\’ll come to moonshine
conjecture of Conway and Norton (for those students who did hear my talk
on Antwerp sprouts, yes both Conway and Simon Norton (via his
SNORT-go) did appear there too…) and if time allows it, I\’ll sketch
the main idea of the proof. Fortunately, Richard Borcherds has written
some excellent expository papers I can use (see his papers-page and I also discovered a beautiful
moonshine-page by Helena Verrill which will make my job a lot
easier.
Btw. yesterday\’s Monster was taken from her other monster story…

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25 years monstrous moonshine

Published April 21, 2004 by lievenlb

Writing a survey paper is a highly underestimated task. I once
tried it out with \’Centers of generic division algebras : the
rationality problem 1965-1990\’ and it took me a lot of time and that
was on a topic with only 10 to 15 key papers to consider… The task of
writing a survey paper on a topic with any breadth must be much more
difficult. Last week, Terry Gannon posted a survey paper on the arXiv :
Monstrous Moonshine : The first twenty-five years
which gives a very readable introduction to this exciting topic. It has
a marvelous opening line :

It has been approximately
twenty-five years since John McKay remarked that

196 884 = 196 883 +
 1

Anyone who is puzzled by this line (“So what?”)
should definitely have a go at this paper! Still not convinced? Here is
the second sentence :

That time has seen the discovery of
important structures, the establishment of another deep connection
between number theory and algebra, and a reinforcement of a new era of
cooperation between pure mathematics and mathematical
physics.

For the remaining sentences (quite a few, the paper
is 33 pages long) I happily refer you to the paper.

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artistic and other frustrations

Published April 17, 2004 by lievenlb

Yesterday, PD1 exhibited some of her paintings in the Antwerp Museum for Photography. Over breakfast this morning she was in a rare angry mood.

No, she was satisfied with the responses she got on her work, the room was not ideal (lighting etc.) but that was not what mattered…

Me : So?

She : There was this other work, a video-performance. I once saw by accident on Arte a short-film and this performance stole the whole idea of that film, from start to finish! The whole idea was nicked!

Me : Wake up! That’s the majority way of creating art, or science for that matter.

She : But it is so unfair! Why do people steal ideas ?

Me : Maybe they don’t see it as stealing. Maybe they believe they do a better thing with the original idea than the person who invented it.

She : Nothing can beat the original! Anyway, I find the most rewarding thing about art to come up with an original idea and work it out. It cannot be rewarding to steal other people\’s ideas.

Me (dry) : I think such people are after other rewards…

She : The same thing happens at school. Sometimes I come up with a suggestion to use a different technique or material and then a few weeks later, half of my class seems
to have worked this out too.

Me : So ? You still had the idea.

She : Yes, but the Jury doesn’t know that!

Me : So ? After the Jury you can still be confident to come up with new ideas, these others may fear they will only be able to repeat themselves.

She : But is it so unfair!

Me : What’s the alternative ? Are you going to lock yourself up in your room to
paint and let nobody see the result?

She : No, but I prefer to do my painting here at home, on my own with nobody looking over my shoulder constantly to see whether they can use some of my ideas. I will
paint on my own and only when it is fully finished they may see the result!

Me : That’s the spirit girl! You are much cleverer than I will ever be…

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