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Explore a 26-minute talk by Gianluca Giacchi from the Workshop on "Quantum Harmonic Analysis" held at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in May 2025. Delve into recent extensions of Hardy's uncertainty principle, a fundamental result in harmonic analysis that establishes limitations on how quickly a function and its Fourier transform can simultaneously decay. Discover how this principle has been expanded to apply to propagators of Schrödinger equations with quadratic Hamiltonians (metaplectic operators), which generalize the Fourier transform and have important applications in time-frequency analysis, quantum harmonic analysis, and signal processing. Learn about concrete examples including fractional Fourier transforms arising from anisotropic harmonic oscillators, and understand how Gaussian decay conditions in specific directions relate to the geometry of corresponding Hamiltonian flows.
