My field is numerical analysis or scientific computing in the broad sense: the study of constructive methods for solving the problems of continuous applied mathematics. My work over the years has been in four particular areas: (1) numerical conformal mapping and its application to fluid mechanics and electrical engineerig; (2) approximation theory and applications, especially approximation by rational functions in real and complex domains; (3) the numerical solution of partial differential equations by finite- difference and spectral methods, especially problems associate with numerical stability and wave propagation; and (4) numerical linear algebra.
My current research centers on a topic that touches both the third and fourth areas: the analysis and algorithmic treatment of non-normal matrices and operators. Non-normality is a generalization of non-symmetry for matrices or non-self-adjointness for operators; it means that the eigenvectors or eigenfunctions are not orthogonal. Although symmetric matrices and self-adjoint operators arewell understood by numerical analysts and applied mathematicians, respectively, we do not have a good understanding of how non-normal matrices and operators behave and how they can best be handled algorithmically. For example, there is no entirely satisfctory analog for nonsymmetric matrices of the conjugate gradient algorithm for symmetric matrices. Unexpected effects arise in highly non-normal problems that have nothing to do with eigenvalues, and I have been developing techniques for understanding suh effects that make use of the norm of the resolvent operator and of a generalization of spectra known as pseudospectra.
I am presently investigating a number of applications of this kind, in numerical methods for partial differential equations, polynomial zerofinding, and fluid mechanics.