Multiparameter harmonic analysis on the Heisenberg group

谐波分析是一个高度活跃的数学领域，最近在语音、图像、心电图以及更一般的数字数据集等信号的分析和处理方面有许多应用。它提供了在不同尺度上展开信号和提取特征的工具，以及压缩信息的方法，这些方法对存储和计算有用。一般来说，谐波分析关注的是将信息分解成更简单的片段。这可以用音乐的模型很好地说明。当拨动调中音C的音叉时，它会产生近乎纯净的音调；也就是说，气压的正弦波变化。另一方面，当像小提琴这样的乐器演奏中音C时，它会产生纯音的总和：在同一基音或基频上以不同强度叠加的是正弦波，其频率是正弦波的倍数。这些被称为高次谐波。正是这些高阶谐波的存在决定了乐器的音色。虽然在两种情况下演奏的是完全相同的音符，但这两种乐器听起来略有不同。人耳在识别声音特征的这种细微差别时，是在进行谐波分析，这种差别是由不同振幅的高谐波所引起的。我的工作主要涉及乘数理论，这是谐波分析中的一个非常重要的工具。它可以用音乐模型来简单地描述。如果我们考虑一些声音或信号，以这种方式被分解成不同幅度的组成频率，假设我们有兴趣阻尼某些频率，也许放大其他频率：我们这样做，例如，在调整立体声音响的均衡器时。乘数理论研究的是这会对整个信号产生什么影响。乘数理论在偏微分方程、解析数论、微分几何等许多数学领域都有重要的应用，甚至在Navier Stokes问题中也发挥了作用，该问题的解现在价值100万美元。

Harmonic analysis is a highly active field of mathematics, which has had many recent applications to the analysis and manipulation of signals such as speech, images, electrocardiograms, as well as more general digital data sets. It provides tools for unraveling signals and extracting features at different scales, as well as methods of compressing information which are useful for storage and computation. Harmonic analysis is concerned, generally speaking, with breaking information up into simpler pieces. This can be nicely illustrated with the model of music. When a tuning fork for middle C is struck, it produces nearly a pure tone; that is, a sine wave variation in the air pressure. On the other hand, when a musical instrument such as a violin plays middle C, it produces a sum of pure tones: superimposed at various intensities on the same ground or fundamental frequency, are sine waves with frequencies which are multiples of this. These are called higher harmonics. It is the presence of these higher harmonics which is responsible for the character, or timbre, of an instrument. Although exactly the same note is being played in the two cases, the two instruments sound slightly different. The human ear, in recognizing this subtle difference in the character of the sound caused by the presence of these higher harmonics at various amplitudes, is doing harmonic analysis. My work largely concerns multiplier theory, which is a very important tool in harmonic analysis. It can be described very simply in the musical model. If we consider some sound, or signal, which has been decomposed in this way into constituent frequencies at various amplitudes, and suppose we are interested in damping certain of those frequencies, and perhaps magnifying others: we do this, for example, when adjusting the equalizer on a stereo set. Multiplier theory is the study of what effect this would have on the signal as a whole. Multiplier theory has important applications in many areas of mathematics, such as partial differential equations, analytic number theory, differential geometry, and even plays a role in the Navier Stokes problem, the solution of which is now worth US\$1m.