# Roadmap to understand the link between spherical harmonics and Riemann sphere?

My ultimate goal is to see how the point of infinity and an arbitrary transform in Riemann sphere can lead to what consequences in dynamical systems, and it seems that harmonic analysis plays a crucial role in between since it connects Fourier transform and spherical harmonics, Hilbert space and functional analysis, topology, group, graph and representation theory in one place. In the term of complex systems theory, this could be the super high-degree node.

What road map should I take? The suggested related topics are (1) Fourier transform of distributions, (2) spectral theorem (spectrum of the Laplacian), (3) harmonic analysis and (4) representations of locally compact groups. I have finished complex analysis (Needham), dynamical systems theory (Strogatz), and the first 6 chapters of Kreyszig's on functional analysis. I have some choices starting from here:

• Keep reading Kreyszig. The rest of the book dedicates to spectral theory and I'm happy to keep reading, but since there are many other things to learn is this the optimized path?
• Switch to Rudin's book. It dedicates two parts specifically for (1) and (2), but it seems to mostly use Banach spaces?
• Switch to harmonic analysis:
• Dym & McKean, Fourier Series and Integrals. But it's too focused on Fourier analysis? It's also pretty old (about 50 years).
• Folland, A Course in Abstract Harmonic Analysis. Seem ideal, but is it too soon to read it now? And it doesn't talk about distributions.

Related questions on harmonic analysis: Undergrad level: Math SE; grad level: Math Overflow, Reddit. Also What's a good primer from linear algebra to spherical harmonics?

• The topics I see related : Fourier transform of distributions, spectral theorem (spectrum of the Laplacian), harmonic analysis and representations of locally compact groups. Not sure what you meant with the Riemann sphere here. – reuns Sep 26 '17 at 19:59
• Under the group of Mobius transformation $PSL(2,\textbf C)$ spherical harmonics can be though of as a Riemann sphere (not sure what this group does). And since the harmonics is the solution of Laplacian equation, I wonder what happens if we apply this equation on the Riemann sphere. – Ooker Sep 27 '17 at 11:21
• The spherical harmonics are an orthonormal basis of $L^2(S^2)$ (where $S^2 = \{ x \in \mathbb{R}^3, \|x\|^2 = 1\}$), obtained by diagonalizing the Laplacian. Now you can map $S^2$ to $\mathbb{C} \cup \infty=\mathbb{P}^1(\mathbb{C})$ the Riemann sphere, having a complex and algebraic structure allowing to define analytic functions. Automorphisms of $\mathbb{P}^1(\mathbb{C})$ are the Möbius transformations (corresponding to rotations of $S^2$, so the Laplacian behaves nicely under those) – reuns Sep 27 '17 at 11:46
• If you care only of simple manifolds such as the sphere, then my comments give most of the keywords (and at first forget about representation theory of locally compact groups). Next time you should give a little more context (some concrete applications -in physics ?- you'd like to study). – reuns Sep 27 '17 at 16:22
• @Perturbative Representation theory "illuminates and generalizes Fourier analysis via harmonic analysis" – Ooker Sep 27 '17 at 21:08