Applications of harmonic analysis to discrete geometry - Machado, Fabrício Caluza (2021)
Harmonic analysis is the analysis of function spaces under the action of some group. In this project we consider applications of Harmonic analysis on Euclidean space, via the group action of translations, and applications of Harmonic analysis on the sphere, via the orthogonal group action. While the analysis on Euclidean space leads to the classical Fourier analysis and operations such as the Fourier transform, representation theory allows us to see the action of the orthogonal group with the same lens, in such a way that to functions of positive type correspond invariant and positive kernels in the sphere and to the Fourier inversion formula corresponds the decomposition of a spherical function into spherical harmonics. In this thesis we apply these elements to three different geometrical problems. In the first project we use semidefinite programming to bound the maximum number of equiangular lines with a fixed common angle in the Euclidean space and we show how this bound relates to previously known bounds for spherical codes and to independent sets in graphs. In the second project we consider the counting of integer points in dilates of a rational polytope P and use the development of the Fourier transform of a polytope via Stokes formula to determine a formula for the second-order Ehrhart coefficient, namely the coefficient of t^(d-2) in | tP intersection Z^d|. In the third project we consider again the Fourier transform of a polytope and use its development via Brion\'s theorem to show that it does not contain circles in its null set. Fourier analysis, polytopes, lattice sums, packing, equiangular lines, semidefinite programming bounds, spherical harmonics, Ehrhart quasi-polynomials.