Terence Tao on Nostr: nprofile1q…wuwy3 The restriction phenomenon, first discovered by Stein, asserts ...

nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpq8syyvsjs4temkau2kwjhujtkpe8r7kemukzpmm8nnulzc8xpkuuq7wuwy3 (nprofile…uwy3) The restriction phenomenon, first discovered by Stein, asserts that the Fourier transform of a function, when restricted to a curved surface such as a sphere or paraboloid, is surprisingly well-behaved in size compared to if one had restricted to a flat plane instead. This phenomenon helps explain the dispersive nature of waves, whose Fourier transform is often concentrated on such surfaces; through geometric optics approximations (the wave packet decomposition), it is also related to geometric incidence problems such as the Kakeya needle problem. In particular, the central conjecture in restriction theory - appropriately named the restriction conjecture - implies one of the major open problems in geometric measure theory, namely the Kakeya conjecture.