Harmonic Analysis and Applications

Jean Bourain的这项数学研究项目集中在与谱理论相关的调和分析问题上。首先要研究的是紧致流形在高能下的本征函数的行为，重点放在平面环面的“最简单模型”上。在这种情况下，本征函数是明确的，但它们的许多性质仍然是猜想的。Bourain打算进一步探索这些本征函数应该遵守的各种矩不等式，这些矩不等式对于薛定谔算子理论中的问题，特别是控制理论中的问题是必不可少的。到目前为止，涉及到的方法很多。在低维中，数论起到了关键作用。来自椭圆曲线理论(2D)、球面上格点的分布性质和Siegel质量公式的输入，带来了值得进一步研究的新见解。在高维方面，最近的进展伴随着振荡积分算子理论的突破。挑战是建立归一化特征函数的高阶矩的一致估计，Bourain最近的一些工作初步接近于这一点。与其他领域的互动使这项研究特别具有启发性。李群中的谱理论提供了一个不同的问题和猜想的全景。Bourain将继续研究光谱间隙，这是中心主题之一，在数论、数学物理和理论计算机科学中有许多应用。Jean Bourgan的这一数学研究项目是在调和分析领域，重点是“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，进化生物学和固体物理学。人们可以列举它与瓦片和准晶体理论、量子计算、纠错码、非均匀介质中的传输等相关的几个方面。Bourain过去与不同的合作者合作的大部分工作都与详细阐述一个能够证明光谱间隙存在的一般框架有关。Bourain还将研究波的相互作用；这个问题是许多来自物理和工程的微分方程解的研究的核心。粗略地说，这些解是由集体效应遵循深奥数学原理的基次谐波叠加而得到的。在许多重要的例子中，例如在薛定谔算子理论中，这一理论仍然远远没有完全建立起来。Bourain将集中讨论关于高能本征态行为的一些主要猜想及其相关方面。其他数学领域也取得了惊人的进展，如动力学和数论，提供了值得进一步探索的新视角。虽然这些进展是不可否认的，几种猜测现象现在也可以证明是合理的，但在已知和鲜为人知的领域仍然存在许多挑战，鲍尔加恩将通过这个项目进行研究。

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.