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Explore advanced geometric inequalities in this graduate-level mathematics lecture from the Instituto de Matemática Pura e Aplicada's 2026 Summer School. Delve into fundamental concepts including mean inequalities, Hölder, Minkowski and Jensen inequalities, Stolarsky means and Palés inequality, and the Khintchine theorem. Study rearrangement inequalities and their applications through Riesz, Chebyshev, and Hardy-Littlewood-Pólya inequalities. Examine the equivalence of majorizations, Schur convexity, symmetric functions and Muirhead's theorem. Investigate Hilbert and Hardy inequalities, Wirtinger's inequality, and Steiner symmetrization. Analyze Hardy-Littlewood-Sobolev inequalities with applications, and conclude with the Brunn-Minkowski inequality and its consequences including isoperimetric and Prékopa-Leindler inequalities. Master these essential tools in geometric analysis through rigorous mathematical exposition based on classical references by Hardy, Littlewood and Pólya, alongside contemporary research in isoperimetric theory and Monge-Ampère equations on spheres.
