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Explore eigenfunctions of the Laplace-Beltrami operator and their relationship to random points in this 46-minute conference talk delivered at the Instituto de Matemática Pura e Aplicada. Delve into advanced mathematical concepts at the intersection of Fourier analysis and spectral geometry as part of the "Fourier Analysis & Beyond" workshop. Learn about the theoretical foundations and applications of eigenfunctions in differential geometry, with particular focus on how these mathematical objects relate to random point distributions. Discover cutting-edge research in harmonic analysis and its connections to other mathematical areas through detailed mathematical exposition and rigorous proofs. Gain insights into modern approaches to studying spectral properties of differential operators on manifolds and their probabilistic interpretations. This presentation forms part of a comprehensive workshop featuring mini-courses and invited talks designed to present emerging topics in Fourier analysis to the mathematical community and stimulate new research directions.
