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Seminários do IMPA

Análise / Equações Diferenciais Parciais

Título |
Uncertainty Principles and Sphere Packings |

Expositor |
Felipe Gonçalves
University of Alberta - Canada |

Data |
Quinta-feira, 24 de maio de 2018, 15:30 |

Local |
Sala 232 |

Resumo |

In this talk we describe a new uncertainty principle for functions: If both $f$ and $\widehat{f}$ are radial and eventually nonnegative, both having nonpositive total mass, then it is impossible for both $f$ and $\widehat{f}$ be nonnegative outside an arbitrarily small neighborhood of the origin. This uncertainty principle is now called +1 uncertainty and it was discovered first by Bourgain, Clozel and Kahane with applications to number theory. However, recently, in joint work with Henry Cohn, we discovered that it possess a twin uncertainty principle, now called $-1$, and that this one is intrinsically connected with the linear programming bounds for sphere packings developed by Cohn and Elkies in early $2000$'s. We were able to solve completely the $+1$ uncertainty in dimension $12$ using the constructions via modular forms from Viazovska solution of the $8$ and $24$ dimensional sphere packing problem (which solves the $-1$ uncertainty in dimensions $8$ and $24$). This talk is aimed for the general public and the relevant background will (tentatively) be given.