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Friday, January 12, 2018
4:30 PM - 5:30 PM
Downs 103

Caltech/UCLA Joint Analysis Seminar

The (Euclidean) Fractal Uncertainty Principle and its proof
Ruixiang Zhang, Mathematics Department, Institute for Advanced Study,
Recently Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which roughly says that: Assuming a function on R R has its Fourier support contained in a fractal set Y Y. Then its L 2 L2 norm on a fractal set X X, whose scale is dual to that of Y Y, cannot be close to its L 2 L2 norm over the whole R R as long as dimX,dimY<1 dim⁡X,dim⁡Y<1. The proof seems quite interesting and unusual to me. I will talk about the ingredients in the proof, including the Beurling-Malliavin Theorem. In the original work of Bourgain and Dyatlov the FUP was ineffective. I will also talk about why we can obtain an effective version (joint work with Long Jin) by looking closely into the proof of the Beurling-Malliavin Theorem and proving a weaker but "more effective" version of it.
For more information, please contact Mathematics Department by phone at 626-395-4335 or by email at [email protected].