Harmonic Analysis and Applications

Jean Bourgain的这个数学研究项目主要研究与谱理论相关的谐波分析问题。研究的第一线是紧致流形在高能量下的特征函数行为，重点是平环面的“最简单模型”。在这种情况下，特征函数是显式的，但它们的许多性质仍然是推测性的。布尔甘打算进一步探索这些特征函数应该服从的各种矩不等式，这些矩不等式对薛定谔算子理论中的问题至关重要，例如，特别是对控制理论。现在涉及到的方法有很多。在低维中，数论起着关键作用。从椭圆曲线理论（二维）、球体上点位的分布特性和西格尔质量公式的输入得到了值得进一步研究的新见解。在高维，最近的进展是与振荡积分算子理论的突破共同取得的。挑战在于如何在归一化特征函数的高阶矩上建立统一的估计，布尔甘最近的一些工作初步接近于此。与其他领域的相互作用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猜想全景。布尔甘将继续研究谱隙，这是数论、数学物理和理论计算机科学中许多应用的中心主题之一。Jean Bourgan的这个数学研究项目是在谐波分析领域，重点关注“谱间隙”的概念：这种概念被认为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。人们可以引用它与贴片和准晶体理论、量子计算、纠错码、非均匀介质中的传输等理论的相关性。布尔甘过去与各种合作者的大部分工作都与阐述一个能够证明谱隙存在的一般框架有关。布尔甘还将研究波的相互作用；这个问题是研究物理学和工程学中许多微分方程解的核心。这些解大致上是由基本谐波的叠加得到的，这些基本谐波的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，这个理论还远远没有完全建立起来。布尔甘将集中讨论高能本征态行为的一些主要猜想及其相关方面。其他数学领域的惊人进展，如动力学和数论，提供了值得进一步探索的新视角。虽然这些进步是不可否认的，一些推测现象现在可以证明是合理的，但在已知和不太为人所知的领域，仍有许多挑战，布尔甘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.