Linear algebra in cubes and biquadrates: arithmetic harmonic analysis

Everybody knows how to solve systems of linear equations. In this talk, we discuss what can be said if all the variables are cubes, or biquadrates (and so on, as Edward Waring once said). This leads us to consider certain exponential sums that are only partly of arithmetic origin. The methods proposed will be distinctly different from the somewhat naive harmonic analysis that had been used hithereto. Almost all original work described here is joint with Trevor Wooley.