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Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to aunit sphere $S^2$, admits essentially a unique representation of the form$\lambda + \sum_{k = 1}^d [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s arelinear forms in $x, y$ and $z$ and $\lambda$ is a real number. The coefficientsof these linear forms, viewed as 3D vectors, are called \emph{multipole}vectors of $P$. In this paper we consider similar multipole representations ofpolynomial and analytic functions on other quadratic surfaces $Q(x, y, z) = c$,real and complex. Over the complex numbers, the above representation is notunique, although the ambiguity is essentially finite. We investigate thecombinatorics that depicts this ambiguity. We link these results with someclassical theorems of harmonic analysis, theorems that describe decompositionsof functions into sums of spherical harmonics. We extend these classicaltheorems (which rely on our understanding of the Laplace operator$\Delta_{S^2}$) to more general differential operators $\Delta_Q$ that areconstructed with the help of the quadratic form $Q(x, y, z)$. Then we introducemodular spaces of multipoles. We study their intricate geometry and topologyusing methods of algebraic geometry and singularity theory. The multipolespaces are ramified over vector or projective spaces, and the compliments tothe ramification sets give rise to a rich family of $K(\pi, 1)$-spaces, where$\pi$ runs over a variety of modified braid groups.

Deaconstructing Functions on Quadratic Surfaces into Multipoles