Partial Differential Equations (PDE)


Our understanding of the fundamental processes of the natural word is based to a large extent on partial differential equations (PDE). For example, the Einstein equations describe the geometry of space-time and its interaction with matter. The dynamics of fluids and elastic solids are governed by partial differential equations that go back to Euler and Cauchy. Electro-magnetic waves including the propagation of light in various media are modeled by Maxwell's equations. More recently, PDE's are gaining importance in the social and life sciences. The Black-Sholes PDE underpins the theory of option pricing in financial mathematics. Reaction-diffusion models are used in neurophysics and population dynamics.

The analysis of PDE's has given rise to new mathematical ideas. For example, the study of the equation thermal diffusion lead to the discovery of Fourier series and ultimately the field of Fourier analysis. Hilbert's investigations into eigenvalue problems arising in the PDE's for vibrating strings and Schrodinger's equation in quantum mechanics evolved into to modern theory of functional analysis and operator theory.

Nonlinear PDE is major research area at UCSB. Bjorn Birnir specializes in application of dynamical systems tools to infinite dimensional systems. Gustavo Ponce is an expert in modern techniques of harmonic analysis applied to nonlinear dispersive equations. Tom Sideris studies nonlinear waves with a particular interest in hydro- and elasto-dynamics. Denis Laubutin studies geometric measure theory in connection with PDE's arising in differential geometry. Close connections exist between the groups in PDE and Applied Mathematics.