H2020Индивидуална стипендия2019–2021

RESTRICTIONAPP · A multilinear approach to the restriction problem with applications to geometric measure theory, the Schrödinger equation and inverse problems

„Хоризонт 2020“ — Действия „Мария Склодовска-Кюри“

Период: 2019-08-01 → 2021-10-02
Финансиране от ЕС: 172 932 €
Участници: 1
Схема: MSCA-IF

Линиите свързват координатора с партньорите.

Накратко на български

Фурие анализът изследва как функциите се разлагат на синуси и косинуси, като тук се фокусира върху ограниченията при повърхности като сфери или конуси. Това помага за по-доброто разбиране на квантовата механика, уравнението на Шрьодингер и предаването на сигнали.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

A multilinear approach to the restriction problem with applications to geometric measure theory, the Schrödinger equation and inverse problems

Fourier analysis originated two centuries ago while trying to understand which periodic functions can be decomposed into a sum of undulatory functions: sines and cosines. The attempt to find the precise necessary and sufficient conditions which guaranteed such a representation was a true motor of developments in mathematics in the nineteenth century; for example, both the Riemann and Lebesgue integration theories and the Cantor set theory originated while trying to understand this representation better. Furthermore, it continues to be used frequently as a tool in science and engineering, from signal transmissions to quantum mechanics. This field continues to be very active and now also considers nonperiodic functions, in which case the sums of undulatory functions are replaced by integrals. We are particularly interested in understanding when we can restrict meaningfully to surfaces such as the cone or the sphere. This subfield of Fourier analysis, called Fourier restriction theory, is of fundamental importance. Many mathematicians, including three Fields Medal awardees, have made recent contributions. A key new tool in Fourier restriction theory are the recently discovered multilinear estimates. The main objective of the project was to further develop the multilinear approach of Fourier restriction theory. Specifically, the project aims to develop the multilinear restriction estimates with sharp dependence on the transversality and apply such estimates to the Schrödinger and wave equations, to inverse problems, as well as to the linear Fourier restriction problem. Despite the project being shortened from two years to ten months (lasting from 01/08/2019 to 31/08/2020 with three months paternity leave), we were able to further understanding of the multilinear estimates in three dimensions by proving a refinement, which will likely be helpful to solve –or at least partially solve– the previously mentioned problems, as well as others.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

The Fourier restriction conjecture, one of main open problems in harmonic analysis, has deep connections with problems in a variety of different fields of mathematics. The aim of this proposal is to further develop the multilinear approach in restriction theory and apply it to several problems in geometric measure theory, the Schrödinger equation and inverse problems. In order to develop this proposal, the Experienced Researcher will join the harmonic analysis group at ICMAT under the supervision of one of its permanent researchers, Keith Rogers, an ERC grant awardee. The host group has extensive experience in the application of harmonic analysis techniques to inverse problems and geometric measure theory, among others. The scientific training strategy of this proposal consists in the assimilation of the techniques of geometric measure theory and inverse problems. While the Researcher is experienced in restriction theory and dispersive equations, as evidenced by his contributions to the field, it is the combination of this prior knowledge with the proposed scientific training that is needed for the successful development of this proposal.This MSC fellowship will achieve a variety of positive outcomes: boosting the convergence of distinct research fields and collaborative networks, producing a synergy with the ERC Starting Grant recently held by the Supervisor, and diversifying the fellow’s mathematical knowledge, ultimately strengthening him as an independent researcher.

Оригинален текст от CORDIS (на английски).

Участници

AGENCIA ESTATAL CONSEJO SUPERIOR DE INVESTIGACIONES CIENTIFICAS · MadridКоординаторИспания