*The 2001 Chern Lectures will be delivered by Peter Lax on February 9-March 12, 2001.*

Department of Mathematics, University of California, Berkeley, presents

The 2001 Chern Lectures

Peter Lax

Professor of Mathematics

Courant Institute, NYU**Review of Dispersive Equations**

Numerical evidence of oscillations and weak convergence. The culprit: too many conservation laws.**Hamiltonian Structure**

Introductory remarks about finite dimensional Hamiltonian systems. The KdV equation as a Hamiltonian system; two Hamiltonian structures. The conservation laws. Solitary waves. The periodic case.**Scattering and Inverse Scattering**

Transmission and reflection coefficients. Eigenvalues and norming constants. Uniqueness theorem. The Gelfand-Levitan-Marchenko equation. Reflectionless potentials. Trace formula.**Solution of KdV via Inverse Scattering**

Linear equations for the reflection coefficients and the norming constants. The Fredholm determinant in the reflectionless case.**The Zero Dispersion Limit**

Asymptotic analysis of direct scattering problem. The minimum problem. The integral equation. Solution of the integral equation in the smooth regime. Solution of the integral equation in the presence of shocks.