This project is mathematical research on linear differential operators. Such operators transform spaces of functions by taking derivatives and multiplying by coefficients. They are often used to represent physical systems. In this situation, their spectrum, a set of numbers determined by how the operator acts on the space, corresponds to possible energy levels of the system. More specifically, Professor Johnson will study linear ordinary differential operators with bounded but non-decaying coefficients (which might be almost periodic, or might be typical paths of a stationary ergodic process). Using techniques from dynamical systems theory, he will study spectral properties and generalized Floquet exponents for such operators. He will apply his results to the following areas among others: bifurcation from invariant tori in nonlinear dynamical systems; the Korteweg -deVries equation with bounded initial data; relations of this to string theory via the Virasoro algebra; the topological index theory of periodic and non-periodic differential equations, and its relation with the homology theory of certain operator algebras.
