Abstract:

In this talk I will review recent advances, joint with Syvert Norsett,
in understanding and implementing methods for quadrature with highly
oscillatory kernels. We develop two methods, one based on an 
asymptotic
expansion and the other on interpolation, that afford very precise
approximation in the presence of high oscillation and critical points,
in one or more dimensions. Time allowing, I will describe some of the
applications of these methods to Fredholm equations of the second kind
and ordinary and partial differential equations with rapidly
oscillating solutions.