Professors Dye and Reich will study several question about the convergence of products of nonexpansive mappings. One of the main questions will be whether the von Neumann algorithm admits a nonlinear extension. Also of fundamental importance is whether von Neumann's conclusion of strong convergence persists when the maps are drawn randomly from more than two projections. Operator theory is that part of mathematics that studies the infinite dimensional generalizations of matrices. In particular, when restricted to finite dimensional subspaces, an operator has the usual linear properties, and thus can be represented by a matrix. The central problem in operator theory is to classify operators satisfying additional conditions given in terms of associated operators (e.g. the adjoint) or in terms of the underlying space. Operator theory underlies much of mathematics, and many of the applications of mathematics to other sciences.
