What do I want to do:- (1) Variable coefficient, elliptic PDE.

[Last revised: 18 February, 2007. Includes corrections noted by Tim Nguyen] What do I want to do:(1) Variable coefficient, elliptic PDE. (2) Monopoles (3) Something else if time permits I have pretty much decided not to use one book but to take bits and pieces from several. For one thing there does not seem to be an approptriate treatment of monopoles. The elliptic part of the course is intended to serve as an introduction to many parts of geometic pde and differential geometry. The part on monopoles is basically an extended application of the first part. First let me recall some things from 18.155 – or at least try to stir some memory of what it is about. Namely, using the Fourier transform we can say a lot about constant coefficient linear differential operators. In this course we want to develop methods to handle variable coefficient, non-linear and (because of gauge-invariance) non-elliptic operators. However in all three directions of generalization we will still ultimaltely use constant coefficient elliptic methods as a starting point. A linear, constant coefficient differential operator can be thought of as a map