We shall discuss various problems of minimizing discrete energies and energy integrals on the sphere (and in other spaces and domains) and many related problems and topics in discrete geometry, such as sums of distances, optimal packing and coverings, spherical designs, discrepancy, Welch bounds, unit norm tight frames, equiangular lines, SIC-POVMs and Zauner's conjecture, p-frame energies, Fejes Toth conjecture on the sum of line angles etc. We shall present both standard approaches to these objects and some more novel points of view through the prism of energy minimization. We shall also discuss numerous relevant tools such as Fourier analysis, spherical harmonics, orthogonal (Chebyshev, Legendre, Gegenbauer) polynomials, positive definiteness, Mercer's theorem in spectral theory, as well as the linear programming method and (if time allows) semidefinite programming.
In this lecture series, I'll describe some open problems in Fourier Analysis that are either not so well-known or forgotten – these problems deserve a good solution, and new ideas are required.