Math 254A - Topics in real analysis

Terence Tao

Required: 245A An oscillatory integral operator is anything of the form

T f(x) = ∫ e^{iλϕ(x,y)}a(x, y)f(y) dy,

where ϕ is a smooth real function, a is some smooth cutoff, the integral is over some manifold, and λ is a real number (usually pretty large). The Fourier transform is a model example of such an operator. In this course we will study these operators, and see why they appear in such diverse topics as

* Convergence of Fourier series

* Probability distributions of eigenvalues of random matrices

* Spectral theory of elliptic operators

* Counting the number of lattice points in a ball

* Scattering of radiation

* Parametrices for the wave equation

* Well-posedness of non-linear evolution equations

* Trying to execute a U-turn using as little space as possible

Many problems are still open in these fields. However, we will not dwell too much on active areas of research, and instead try to give a broad survey in these areas. The aim is to prove one modest but non-trivial result in each area.

A basic understanding of the Fourier transform and some real analysis is assumed (Royden will do). We won’t be following any particular book - no single book covers all the above stuff anyway - but we’ll return to Stein’s ”Harmonic Analysis” more often than most other books.

I’ll be at a conference on Monday, Jan 11, so the first class will start on Wednesday, Jan 13.