Harmonic Analysis and Applications

这个数学研究项目由Jean Bourgain专注于与谱理论相关的谐波分析问题。 研究的第一条线与高能量下紧致流形的本征函数的行为有关，重点是平坦环面的“最简单模型”。在这种情况下，本征函数是显式的，但它们的许多性质仍然是显式的。 布尔甘打算进一步探讨各种时刻的不平等，这些本征函数应该遵守和这是必不可少的问题，理论的薛定谔算子，例如，特别是控制理论。 到目前为止，所涉及的方法很多。 在低维中，数论起着关键作用。 椭圆曲线理论（二维）的输入，球体上格点的分布特性和Siegel的质量公式导致了值得进一步研究的新见解。 在高维空间中，近年来的进展与振荡积分算子理论的突破同时出现。 面临的挑战是建立统一的估计高阶矩的归一化本征函数和一些布尔甘最近的工作来试探性地接近这一点。 与其他领域的互动使这项研究特别令人兴奋。 李群中的谱理论提供了一个不同的问题和全景图。 布尔甘将继续研究光谱差距，其中一个中心主题与许多应用数论，数学物理和理论计算机科学。 Jean Bourgan的这个数学研究项目是谐波分析领域的，重点是“光谱间隙”的概念：这种概念具有广泛的跨学科意义，从纯数学到计算机科学，进化生物学和固态物理学。人们可以引用它的相关性理论的平铺和准晶体，量子计算，纠错码，运输在非均匀介质中，仅举几例。 Bourgain过去与各种合作者的大部分工作都与制定一个通用框架有关，该框架能够证明光谱间隙的存在。 布尔甘还将调查波的相互作用;这个问题的核心在于研究的解决方案，许多微分方程从物理和工程。 这些解大致上是由基本谐波叠加而得，其集体效应遵循深刻的数学原理。 在许多重要的例子中，例如在薛定谔算子理论中，这个理论还远未完全建立。 Bourgain将集中在一些主要的本征态的行为在高能量和相关方面的知识。 其他数学领域，如动力学和数论，也取得了惊人的进展，提供了值得进一步探索的新视角。虽然这些进步是不可否认的，现在可以证明几种自然现象是合理的，但在已知和不太为人所知的领域仍然存在许多挑战，Bourgain将通过这个项目进行调查。

This mathematics research project by Jean Bourgain is focused on harmonic analysis problems related to spectral theory. A first line of research has to do with the behavior of eigenfunctions of compact manifolds at high energy, with focus on the "simplest model" of the flat tori. In this setting, the eigenfunctions are explicit but nevertheless many of their properties remain conjectural. Bourgain intends to explore further the various moment inequalities which these eigenfunctions are supposed to obey and which are essential to issues in the theory of Schrodinger operators for instance, in particular to control theory. There is by now a large array of methods involved. In low dimension, number theory plays a key role. Input from elliptic curve theory (in 2D), distributional properties of lattice points on spheres and Siegel's mass formula led to new insights that deserve further study. In high dimension, recent progress came jointly with breakthroughs in the theory of oscillatory integral operators. The challenge is to establish uniform estimates on higher moments of the normalized eigenfunctions and some of Bourgain's recent work comes tentatively close to this. The interaction with other fields makes this research particularly stimulating. Spectral theory in Lie groups offers a different problematic and panorama of conjectures. Bourgain will continue research on spectral gaps, one of the central themes with many applications to number theory, mathematical physics and theoretical computer science. This mathematics research project by Jean Bourgan is in the area of harmonic analysis with a focus on the notion of "spectral gap": such notion is known to have broad inter-disciplinary significance, ranging from pure mathematics to computer science, evolutionary biology and solid state physics. One can cite its relevance to the theory of tilings and quasi-crystals, quantum computation, error correcting codes, transport in inhomogeneous media, to mention a few. Much of Bourgain's past work with various collaborators has to do with elaborating a general framework enabling to prove the existence of spectral gaps. Bourgain will also investigate wave interactions; this problem lies at the heart of the study of solutions of many differential equations from physics and engineering. These solutions are roughly speaking obtained by superposition of elementary harmonics which collective effect obey deep mathematical principles. In many important examples, for instance in the theory of Schrodinger operators, this theory is still far from completely established. Bourgain will focus on some of the main conjectures on the behavior of eigenstates at high energy and the related aspects. Striking advances came from other mathematical areas, such as dynamics and number theory, offering new perspective that deserve further exploration. While the advances are undeniable and several conjectural phenomena can now be justified, there remain many challenges in known and less known territory which Bourgain will investigate through this project.