Postcards from the edge, or Snapshots of the theory of generalised Moonshine

We provide a physics derivation of Monstrous moonshine. We show that the McKay-Thompson series $T_g$, $g\in \mathbb{M}$, can be interpreted as supersymmetric indices counting spacetime BPS-states in certain heterotic string models. The invariance groups of these series arise naturally as spacetime T-duality groups and their genus zero property descends from the behaviour of these heterotic models in suitable decompactification limits. We also show that the space of BPS-states...

A homage to the life and mathematics of John K. S. McKay. Obituary for the Bulletin of the London Mathematical Society.

These informal notes, initially prepared a few years ago, look at various questions related to infinite processes in several parts of mathematics, with emphasis on examples.

The Conway--Norton conjectures unexpectedly related the Monster with certain special modular functions (Hauptmoduls). Their proof by Borcherds et al was remarkable for demonstrating the rich mathematics implicit there. Unfortunately Moonshine remained almost as mysterious after the proof as before. In particular, a computer check - as opposed to a general conceptual argument - was used to verify the Monster functions equal the appropriate modular functions. This, the so-calle...

Lecture given Thursday 22 October 1992 at a Mathematics-Computer Science Colloquium at the University of New Mexico. The lecture was videotaped; this is an edited transcript.

This article covers my second talk at the Gathering for Gardner in March, 2010. It is about an Odd One Out puzzle I invented, after having been inspired by Martin Gardner. I do not like Odd One Out questions; that is why I invented one.

This document contains a description of several of my papers, including remarks on history and connection with subsequent work. It also contains some new results and conjectures.

The mathematical study of infinity seems to have the ability to transport the mind to lofty and unusual realms. Decades ago, I was transported in this way by Rudy Rucker's book Infinity and the Mind. Despite much subsequent learning and teaching of mathematics in the service of physics and astronomy, there remained quite a few aspects of the "higher infinities" that I was still far from comprehending. Thus, I wanted to dive back in to understand those ideas and to find good w...

There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been sleep-walking in their infinitary mathematical paradise to take heed. Much of mathematics relies upon either (i) the "existence'" of objects that contain an infinite number of elements, (ii) our ability, "in theory", to compute with an arb...

This is a non-standard paper, containing some problems, mainly in model theory, which I have, in various degrees, been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discus...