
a monstrous unimodular lattice
An integral $n$dimensional lattice $L$ is the set of all integral linear combinations \[ L = \mathbb{Z} \lambda_1 \oplus \dots \oplus \mathbb{Z} \lambda_n \] of base vectors $\{ \lambda_1,\dots,\lambda_n \}$ of $\mathbb{R}^n$, equipped with the usual (positive definite) inner product, satisfying \[ (\lambda, \mu ) \in \mathbb{Z} \quad \text{for all $\lambda,\mu \in \mathbb{Z}$.} \] But […]

Borcherds’ favourite numbers
Whenever I visit someone’s YouTube or Twitter profile page, I hope to see an interesting banner image. Here’s the one from Richard Borcherds’ YouTube Channel. Not too surprisingly for Borcherds, almost all of these numbers are related to the monster group or its moonshine. Let’s try to decode them, in no particular order. 196884 John…

What we (don’t) know
Do we know why the monster exists and why there’s moonshine around it? The answer depends on whether or not you believe that vertex operator algebras are natural, elegant and inescapable objects. the monster Simple groups often arise from symmetries of exceptionally nice mathematical objects. The smallest of them all, $A_5$, gives us the rotation…

a noncommutative Jack Daniels problem
At a seminar at the College de France in 1975, Tits wrote down the order of the monster group \[ \# \mathbb{M} = 2^{46}.3^{20}.5^9.7^6.11^2.13^3.17·19·23·29·31·41·47·59·71 \] Andrew Ogg, who attended the talk, noticed that the prime divisors are precisely the primes $p$ for which the characteristic $p$ supersingular $j$invariants are all defined over $\mathbb{F}_p$. Here’s Ogg’s…

the McKayThompson series
Monstrous moonshine was born (sometime in 1978) the moment John McKay realized that the linear term in the jfunction $j(q) = \frac{1}{q} + 744 + 196884 q + 21493760 q^2 + 864229970 q^3 + \ldots $ is surprisingly close to the dimension of the smallest nontrivial irreducible representation of the monster group, which is 196883.…

thanks for linking
I’ve reinstalled the Google analytics plugin on december 22nd, so it is harvesting data for three weeks only. Still, it is an interesting tool to gain insight in the social networking aspect of mathblogging, something I’m still very bad at… Below the list of all blogs referring at least 10 times over this last three…