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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2003-2004 MATHEMATICS COLLOQUIUM SERIES 02/19/03 Colloquium DR. OLEG KOVRIJKINE Massachusetts Institute of Technology The Uncertainty Principle in Fourier Analysis   The Uncertainty Principle is the following vague statement: a function and its Fourier transform can not both be concentrated on small sets. The famous Heysenberg Uncertainty Principle in Quantum Mechanics is a particular version of this general principle. We consider the following type of the Uncertainty Principle:  where C does not depend on  and E and  are “small” sets in and, in particular, it implies that if f is supported on E and if is supported on then . The challenging part is to find such pair of sets E and . There are three main results of this type: the Amrein-Berthier theorem where E and  are so called -thin sets. The last two theorem have numerous applications for PDE. We obtain a new version of the Uncertainty Principle which links the last two theorems by introducing a new notion of density for sets E and . The obtained result is sharp. We apply the new result to estimate certain operators on .