(joint work with Lam, Moeller and Shimakura) If V is a holomorphic vertex algebra of central charge 24 then its weight one space V_1 is known to be a reductive Lie algebra which is either trivial, abelian of dimension 24 (in which case V is the Leech lattice vertex algebra) or else one of 69 semisimple Lie algebras first determined by Schellekens in 1993. Until now the only known proof of Schelekens result was a heavily computational one involving case analysis and difficult integer programming problems. Recently Moeller and Scheithauer have established a bound on the dimension of the weight one space of a holomorphic orbifold vertex algebra, using the Deligne bound on the growth of coefficients of weight 2 cusp forms. In this talk I will describe how the dimension bound together with Kac's very strange formula implies that all holomorphic vertex algebras of central charge 24 and nontrivial weight one space are orbifolds of the Leech lattice algebra. Since the automorphism group of the latter algebra is known one can, with a little more work, recover Schellekens result in this way.
We all know the story of how Calculus started as a non-rigorous confusion of intuitive notions of the infinitely small and infinitely big and how eventually the field was put on rigorous footing with the invention of the limit. This story leaves out an interesting question: If the foundation using infinitesimal numbers was so treacherous, why is it that so many of the results obtained using this faulty foundation continued to be true when using the limit formulation, a formulation where all the proofs look different? In this talk, we will explore a modification of the Cauchy construction of the reals that results in an ordered field very closely resembling the real numbers except that it has infinitesimal and infinite elements (and so is not Archimedean). If time permits, we will talk about how some of the basics of Calculus can be formulated in this larger field.