Session S02 - Diverse Aspects of Elliptic PDEs and Related Problems

Thursday, July 15, 19:30 ~ 20:00 UTC-3

Uncertainty principles and interpolation formulae in analysis and PDE

João Pedro Gonçalves Ramos

ETH Zürich, Switzerland   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The famous Heisenberg uncertainty principles predicts, in one of its versions, that a functions $f$ and its Fourier transform $\widehat{f}$ cannot both be too concentrated in space, otherwise $f \equiv 0.$ This intriguing principle has many classical counterparts of similar flavour, such as the Hardy uncertainty principle and the Amrein-Berthier theorem on annihilating pairs.

In recent years, however, several breakthrough results have shed new light onto problems in the uncertainty realm. In particular, we mention first, in the realm of PDEs, a series of papers by Escauriaza-Kenig-Ponce-Vega, which generalised the Hardy uncertainty to a more general Schrödinger equation setting in its sharp form. Secondly, in the purely analytics realm, we mention the Radchenko-Viazovska interpolation formula, which provides one with an explicit way to recover an even function $f \in \mathcal{S}(\mathbb{R})$ given its values $f(\sqrt{n}), \widehat{f}(\sqrt{n}), n \ge 0.$

In this talk, we will go through some of these uncertainty results, from the most classical to some of the most recent, mentioning possible directions, open problems and conjectures in this area. In particular, we shall emphasise more ideas than technical proofs, making this talk an invitation to contribute in the subject.

Joint work with Mateus Sousa (BCAM, Bilbao, Spain), Christoph Kehle (ETH Zürich, Switzerland) and Martin Stoller (EPFL, Switzerland).

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