Invited Presentation 12 Chair: Gregory Kriegsmann, New Jersey Institute of Technology
Evolution Processes: Linear and Nonlinear, Normal and Non-normal, Stable and Unstable Is the solar system stable? Why does the earth's magnetic field oscillate erratically? Can the Lax-Wendroff formula be used to solve the Korteweg-de Vries equation? These are examples of the kinds of questions involving evolution processes that arise throughout the mathematical sciences. In the past two decades such problems have acquired new life through the infusion of ideas from dynamical systems and a heightened recognition of the importance of nonlinearity.
For some years the speaker and a number of colleagues have been involved with the study of another aspect of the behavior of evolution processes, namely, effects that are essentially linear but non-normal, meaning that the linearized evolution operator has non-orthogonal eigenfunctions. Like nonlinearity, non-normality leads to some surprising and counter-intuitive phenomena. Studying such phenomena has generated new insight into the subtle interplays between linear and nonlinear mechanisms and between short-time and long-time behavior.
The speaker will present a survey of some of the results that have been developed in this area, giving special attention to one of the oldest problems of applied mathematics, which has proven to be also one of the strangest: hydrodynamic stability of flow through a pipe.
Lloyd N. Trefethen, Department of Computer Science and Center for Applied Mathematics, Cornell University
Nick Trefethen is a numerical analyst with varied interests, including linear algebra, partial differential equations, approximation theory, conformal mapping, and fluid mechanics. He received his Ph.D. from the numerical analysis group at Stanford (1982), then served as a postdoc at the Courant Institute (1982-1984) and as a faculty member in Applied Mathematics at MIT (1984-1991). He is currently Professor of Computer Science at Cornell University and also a member of Cornell's Center for Applied Mathematics.
Dr. Trefethen was the winner of the first Fox Prize in Numerical Analysis (1985), and has been a Presidential Young Investigator (1986-1991). The theme of his SIAM talk will be elaborated in a book soon to be completed, "Spectra and Pseudospectra: The Behavior of Non-Normal Matrices and Operators."