Explore a 54-minute lecture on sufficient geometric conditions for the null-controllability of evolution equations with Gelfand-Shilov smoothing properties. Delve into the relationship between Gelfand-Shilov regularity indices and control subset geometry for ensuring null-controllability in positive time. Examine new uncertainty principles for finite combinations of Hermite functions, with explicit constant control relative to harmonic oscillator eigenvalues. Discover applications to null-controllability in fractional harmonic oscillators and hypoelliptic non-selfadjoint quadratic equations. Presented by Karel Pravda-Starov as part of the Hausdorff Trimester Program on Kinetic Theory at the Hausdorff Center for Mathematics, this lecture offers insights into advanced mathematical concepts in evolution equations and control theory.