250044 VO Fourier Methods on Manifolds with Applications (2019W)

Fourier methods have been proven very powerful in many areas of pure and applied mathematics. In particular, the Fast Fourier Transform (FFT) builds a very effective bridge between the rich field of Fourier analysis and numerical implementations. Applications embrace for instance signal/image processing, tomographic reconstruction, solving of partial differential equations, simulation of dynamical systems.

The goal of the course is to provide an insight in the range of applications where Fourier methods are particular fruitful. The first part pays attention to classical Fourier analysis of periodic functions (on the circle). After recalling some fundamentals of Fourier series and function spaces, we shall get a first glimpse how Fourier methods can efficiently solve selected problems. Afterwards we like to generalize the concepts to Riemannian manifolds, including examples of recent research.

Depending on the knowledge or preference of the audience, we may focus on specific topics from harmonic analysis, geometry, optimization or numerical implementation.