Harmonic analysis is a branch of analysis in which the action of a topological group plays an essential role. Often seen as a generalisation of Fourier analysis, this area of mathematics provides us with tools to simplify and solve certain families of PDEs.

In this talk we will focus mainly on torus bundles over $S^1$, e.g. the Heisenberg manifold. First making use of the Fourier transform to decompose the space of $L^2$ functions and then viewing the same construction from a representation theory perspective. We will also see how these techniques can be applied to a range of PDEs which would otherwise be quite tricky.

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