Decouplings and applications

首席研究人员与Jean Bourain合作，最近创造了一套新的工具，可以成功地解决数论和偏微分方程领域的一系列问题。直到最近，这些问题中的许多似乎都是无关的。鉴于他与布尔格纳的合作，这些问题现在被理解为两人所称的“脱钩”的更一般理论的一部分。这些方法属于现代调和分析领域，这是一种自然的框架，允许提出足够普遍的理论。该项目旨在扩大解耦的适用范围，并有一些高价值的目标在望。这项研究的一个令人惊讶的特点是，它取消了对频率的某些限制，这些限制在早期的研究中被认为是必要的。特别是，频率具有整数坐标的旧要求被较弱的假设所取代，即频率之间存在足够的空间间隔。预计将开发的工具将对大部分数学界可用和有用。解耦是Littlewood-Paley理论在曲率存在的情况下的某些推广。到目前为止，首席研究员在追求与这一主题相关的研究路线方面的进展依赖于多线性和线性多尺度分析之间的相互作用。他成功地解决了当相关流形是具有非零高斯曲率的超曲面的情况。他现在建议发展非退化曲线的最优解耦理论。这种理论有可能实现调和分析在数论中几乎史无前例的应用。一个值得注意的例子是维诺格拉多夫中值定理的解决。对于圆锥体，有一系列有趣的相关问题。它具有零高斯曲率的事实带来了一个新的难度水平，这肯定需要新的想法。了解圆锥体是一个更雄心勃勃的项目的一部分，该项目旨在了解真实解析曲面的解耦理论。关于各种限制定理和Kakeya猜想，还有更重要的相关问题有待探索。

The principal investigator, in collaboration with Jean Bourgain, has recently created a new set of tools that can successfully address a wide range of problems in the fields of number theory and partial differential equations. Until recently, many of these problems seemed unrelated. In light of his work with Bourgain, these problems are now understood as part of a more general theory that the two call "decoupling." The methods pertain to the field of modern harmonic analysis, a natural framework that allows for the formulation of a general enough theory. This project seeks to enlarge the range of applicability of decouplings, with some high-value targets in sight. A surprising feature of the research is that it removes certain restrictions on frequencies that were thought to be necessary in earlier work. In particular, the old requirement that frequencies have integer coordinates is replaced with the weaker assumption that sufficient spatial separation exists between frequencies. It is expected that the tools that will be developed will be accessible and useful to a large part of the mathematical community.Decouplings are certain generalizations of the Littlewood--Paley theory in the presence of curvature. The principal investigator's progress in pursuing the line of research related to this subject has relied hitherto on the interplay between multilinear and linear multiscale analysis. He has successfully addressed the case when the relevant manifold is a hypersurface with nonzero Gaussian curvature. He now proposes to develop the optimal decoupling theory for nondegenerate curves. Such a theory has the potential to achieve almost unprecedented applications of harmonic analysis to number theory. One notable example is the resolution of Vinogradov's mean value theorem. There is an interesting related circle of problems for the cone. The fact that it has zero Gaussian curvature poses a new level of difficulty that will most certainly require new ideas. Understanding the cone is part of a more ambitious project that will aim at understanding the decoupling theory for real analytic surfaces. There are further important related questions that remain to be explored, in connection with various restriction theorems and the Kakeya conjectures.