Tag: Borcherds

  • 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 […]

  • Teapot supremacy

    No, this is not another timely post about the British Royal family. It’s about Richard Borcherds’ “teapot test” for quantum computers. A lot of money is being thrown at the quantum computing hype, causing people to leave academia for quantum computing firms. A recent example (hitting the press even in Belgium) being the move of…

  • 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…

  • Richard Borcherds on Witt and the Leech lattice

    A rare benefit of the Covid-situation is that Richard Borcherds decided to set up a YouTube channel with recordings of his online lectures. Plenty of gems to be discovered there, including a talk on Monstrous Moonshine, and a talk he gave for the Archimedeans about the Sporadic Groups. As part of his History of Science-course…

  • 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 non-commutative 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 $p$ super-singular $j$-invariants are all defined over $\mathbb{F}_p$. Here’s Ogg’s…

  • Dedekind or Klein ?

    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 try…

  • the McKay-Thompson series

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

  • thanks for linking

    I’ve re-installed 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 math-blogging, something I’m still very bad at… Below the list of all blogs referring at least 10 times over this last three…

  • symmetry and the monster

    Mark Ronan has written a beautiful book intended for the general public on Symmetry and the Monster. The book’s main theme is the classification of the finite simple groups. It starts off with the introduction of groups by Galois, gives the classifivcation of the finite Lie groups, the Feit-Thompson theorem and the construction of several…