MSI Weekly Bulletin - Week starting Monday 28 November, 2005

We discuss some theoretical aspects of solving high-dimensional elliptic PDEs (resulting eg. from homogenization, quantum chemistry, mathematical finance etc.) by adaptive tensor product (sparse grid) methods. We address approximation theoretical, regularity and algorithmic issues.

In this talk we will review the definition and examples of vertex algebras. These algebras were introduced by Borcherds in his work on the Monstrous Moonshine, for which Borcherds was awarded a Fields Medal. We will sketch the construction of the vertex algebra which has the Monster sporadic simple group as the group of automorphisms, and which character is given by the modular function j. Vertex algebras have applications in several areas of mathematics and physics. These objects play a crucial role in constructing Conformal Field Theories, they are connected to the theory of modular forms, and have applications to the soliton PDEs. Vertex algebras provide efficient tools for constructing representations of infinite-dimensional Lie algebras. I will illustrate this with some of the methods used in my work on representations of toroidal Lie algebras.