Harmonic Analysis and Geometric Partial Differential Equations

Nahmod的研究是谐波分析、几何和偏微分方程的交叉。它的目的是研究几何、铁磁和规范场理论中产生的非线性波的行为；以及沿积分曲线曲率不足的向量场的函数。第一部分的重点是几何偏微分方程。特别有趣的是薛定谔图、波图和其他规范场理论，如闵可夫斯基时空中的杨密尔方程。所有这些方程都是“波状现象”的模型。它们的解是相应能量泛函的最小化。为了符合自然物理情况，研究它们在最小正则性假设下的存在性、唯一性是很有意义的。这些都是很难的问题因为这些方程的非线性不仅涉及到解还涉及到它们的导数。Nahmod将解决这些问题，并计划证明，在尺度不变的情况下，只要相对于临界正则范数测量的数据足够小，柯西初值问题的解就存在全局。她还计划解决稳定问题；例如，当数据能量较小时，随着时间的推移，系统是否保持接近其初始状态。后者从物理的观点来模拟这些系统是否接近平衡。这些技术利用这些方程的几何方面来提取关键信息——比如非线性中的特殊结构——然后用于分析。该方法将深度傅立叶分析与规范几何工具相结合。第二部分的目标是研究二维中沿向量场的希尔伯特变换及其相关的极大算子。他们的处理方法与经典的奇异积分研究不同，因为在目前的情况下，奇点存在于一些在每一点都在变化的变化中。Nahmod将研究如何发展时频技术来研究无曲率假设下的算子。例如，在研究函数沿向量场的可微性时就是这种情况。偏微分方程是物理世界中支配许多现象的规律的数学模型。波动方程模拟了不同种类的波（如光波）在均匀介质中的传播。在量子力学中出现了保守型的非线性模型，而在振动系统和半导体的研究中也出现了其他的变体。非线性薛定谔方程出现在描述非线性波的各种物理环境中，例如激光束在折射率对波幅敏感的介质中的传播，理想流体自由表面的水波以及等离子体波。其中一些有趣的问题是关于解的局部存在性和全局存在性、全局解的唯一性以及全局解的长时间行为。数学分析的作用是理解这些方程的解的行为，提供提取其定量和定性信息的工具，并为开发准确近似解的方法奠定基础。傅里叶分析和更广义的频率分解（如时频分析）包括通过“调制波形”将复杂对象分解为局部化且易于理解的基本构建块，然后以直接的方式将它们拼凑在一起。它的工作原理与乐谱非常相似。调制波形有四个属性：振幅（响度），尺度（持续时间），频率（音高）和位置（播放的瞬间）。对象可以是语音、雷达信号，也可以是光学中的振荡表达式、交流驱逐散射、波传播和其他非局域性质的现象。

Nahmod's research lies in the overlap of harmonic analysis, geometry and partial differential equations. It aims at studying the behavior of nonlinear waves arising in geometry, ferromagnetism and gauge field theories; and that of functions along vector fields whose integral curves lack sufficient curvature. In the first part the focus is in geometric partial differential equations. Of special interest are Schroedinger maps, Wave maps and other gauge field theories such as the Yang Mills equationsin Minkowski space-time. All of these equations model `wave like phenomena'. Their solutions arise as minimizers of the corresponding energy functionals. To conform with natural physical situations, it is of interest to study their existence, uniqueness under minimal regularity assumptions. These are difficult issues because the nonlinearities of these equations involve not just the solutions but also their derivatives. Nahmod will address these questions and plans to show that in the scale invariant set up solutions to the Cauchy initial value problem exist globally provided that the data is sufficiently small when measured relative to the critical regularity norm. She also plans stability issues; e.g. whether such a system remains close to its initial state as time evolves when the data has small energy. From a physical viewpoint the latter models whether such systems are close to equilibrium. The techniques exploit geometric aspects of these equations to extract crucial information -such as special structures in the nonlinearity- which is then used in the analysis. The method combines deep Fourier analysis with gauge theoretic geometric tools. The goal of the second part is the study of the Hilbert transform along vector fields and its associated maximal operator in two dimensions. Their treatment departs from the classical study of singular integrals for in the present situation, the singularity lives on some variety that is changing at each point. Nahmod will investigate how to develop time frequency techniques to study operators under no curvature assumptions. This is the case, for example, in studying differentiability properties of functions along vector fields. Partial differential equations are the mathematical models to the laws governing much of the phenomena in our physical world. The wave equation models the propagation of different kind of waves -such as light waves- in homogeneous media. Nonlinear models of conservative type arise in quantum mechanics while other variants appear for example in the study of vibrating systems and semiconductors. The nonlinear Schroedinger equation arises in various physical contexts in the description of nonlinear waves- such as propagation of a laser beam in a medium whose index of refraction is sensitive to the wave amplitude, water waves at the free surface of an ideal fluid as well as in plasma waves. Some of the interesting questions are those about local and global existence of solutions, uniqueness as well as long time behavior of global solutions. The role of mathematical analysis is to understand the behavior of the solutions to these equations, provide the tools to extract their quantitative and qualitative information and lay the foundations upon which methods to accurately approximate the solutions are developed. Fourier analysis and more generalized adapted frequency decompositions such as time-frequency analysis' consists in decomposing complex objects via `modulated waveforms' into basic building blocks which are localized and easy to understand, and then piecing them back together in a straightforward manner. It works very similarly to a musical score. The modulated waveforms have four attributes: amplitude (loudness), scale (duration), frequency (pitch) and position (instant it is played). The objects could be speech, radar signals, as well as oscillatory expressions arising in optics, AC ousting scattering, wave propagation and other phenomena of nonlocal nature.