Applied Math Colloquium

Stephen Wolfram will describe ideas and discoveries from his book "A New Kind of Science," their implications for various fields of science, and their personal and historical context.

The theme of this presentation is eigenspace computation, an ubiquitous task in engineering and physical sciences. More precisely, we consider the problem of iteratively refining estimates of an eigenspace of an n-by-n data matrix. This problem nicely lends itself to a geometric approach, since the iterates belong to a particular non-Euclidean set, the set of fixed-dimensional subspaces of Rn, commonly referred to as the Grassmann manifold.

In order to derive numerical algorithms from the geometric approach, it is essential to understand the relation between the geometric objects (i.e. subspaces) and the matrix representations of these entities. These matrix representations of subspaces admit a topological structure of GL-principal fiber bundle over the Grassmann manifold. Moreover, with a suitable choice of metrics, the bundle projection can be turned into a Riemannian submersion.

We take advantage of this geometric structure to derive a Newton method that relies on the Riemannian geometry of the Grassmann manifold. It converges locally quadratically to the eigenspaces of the data matrix, and even cubically when the data matrix is symmetric.

This method is closely related to several available algorithms for eigenspace computation.This presentation builds on collaborations with Rodolphe Sepulchre (University of Liege, Belgium), Robert Mahony (Australian National University), Paul Van Dooren (Universite Catholique de Louvain, Belgium) and Kyle Gallivan (Florida State University).

Quantum gravity can be Taylor expanded in powers of E^2 G, where E is the energy scale of the interactions considered, and G is Newton's constant in units where \hbar = c = 1. All terms are well-defined, except for the fact that, at every order, new, freely adjustable parameters appear. This means that gravity does not require deviations from standard quantummechanical procedures at any finite order in E^2 G. If, on the other hand, we wish a non-perturbative treatment, in cases where E^2 G >~ 1, fundamentally new approaches are needed. My consistency requirements are more stringent than what presently can be offered by superstring theory and its relatives, supergravity and M-theory. The issue of the interpretation of Quantum Mechanics cannot be avoided.

Modern methods to analyze the asymptotic distribution of the singular values of large random matrices have found applications in the analysis of the Shannon capacity and other limits of wireless communication channels. I will give a tutorial overview of those methods and applications.