Explore sharp inequalities in probability and harmonic analysis through this 59-minute AMS Invited Address given by Rodrigo Bañuelos from Purdue University at the 2023 Joint Mathematics Meetings. Delve into motivations, isoparametric inequality, and the historical context of the problem, including contributions from David Hilbert and Mary Cartwright. Examine proofs involving discrete singular integrals and discover applications in stochastic integrals, exponential sums, operators on lattices, and numerical approximations. Gain insights into p-integrability and conjugate functions, and engage with thought-provoking questions in this comprehensive mathematical exploration.