Symmetry Parameter Analysis of Singular Integrals

谐波分析是有关信号（函数）及其处理（算子）的严格描述的数学分支。信号的例子有声音、图像、时间序列和天气数据。这样的信号是通过叠加（叠加）的基本谐波的明确规定的持续时间，强度和频率。这些基本的谐波被称为小波。图像或音频去噪、压缩或模式识别都是通过滤波器处理来完成的，这是指在滤波器作用于每个小波之后对其进行适当的叠加。这也被称为时频法。重建过程的一个具体例子是在层析成像中使用的，其中固体的形状是由物体的样本沿着穿透波的一维或二维射线重新组合而成的，这些射线可以在数学上描述为三维空间中的线或面。该数学研究项目的一个组成部分侧重于一类奇异积分算子的小波描述的新方法，例如在高振荡信号的时频分析中出现。这个研究项目的另一个组成部分是关于沿直线或平面取样固体物体的数学性质。综合的更广泛影响活动侧重于加强社会经济上处于不利地位的少数民族学生（代表性不足的群体）获得数学研究生学位和提高留校率。还将开展与分析和专题知识传播方面的研究生培训和指导有关的活动。第一个问题的主要目的是根据保持相同不变结构的所谓模型算子，给出奇异积分类的表示公式。这种范式适用于zygmund型算子和类似于双线性Hilbert变换的调制不变算子，这两种算子都是二元概率方法无法达到的。具体应用包括椭圆和色散偏微分方程、算子理论和拟保角映射。第二个相关的问题是由banach值函数的双线性遍历平均的逐点收敛性引起的，这是布尔甘在标量情况下的一个著名定理。该方法基于截断双线性希尔伯特变换的banach值变分估计。关于方向奇异积分的进一步问题的中心项目是Kakeya极大估计的一个版本，其中管状平均值被奇异线段上的平均值所取代，更一般地说，n维子空间。动机的一个来源是与高协维的傅里叶限制的联系。这些方法涉及代数几何技术，如流形上的多项式分划。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Harmonic Analysis is the branch of mathematics concerned with the rigorous description of signals (functions) and of their processing (operators). Examples of signals are sound, images, time series and weather data. Such signals are analyzed via the overlaying (superposition) of basic harmonics of well-specified duration, intensity and frequency. These basic harmonics are functions called wavelets. Image or audio denoising, compression, or pattern recognition are accomplished by filter processing, which refers to a suitable superposition of each wavelet after the action of the filter on it. This is also known as the time-frequency method. A particular concrete example of a re-construction process is used in tomographic imaging, where the shape of a solid body is re-composed from samples of the body along one or two-dimensional rays of penetrating waves, which can be mathematically described as lines or planes in three dimensional space. One component of this mathematics research project focuses on a new family of methods for the wavelet description of the class of singular integral operators, arising for instance in the time-frequency analysis of highly oscillatory signals. Another component of this research project is concerned with the mathematical properties of sampling solid objects along lines or planes. The integrated broader impact activities focus on strengthening the pool of socioeconomically disadvantaged, ethnical minority students (underrepresented groups) in graduate degrees in mathematics and improving retention. Activities connected to training and mentoring of graduate students in Analysis and topical dissemination of knowledge will also be carried out. The broad aim of the first circle of questions is to produce representation formulas for classes of singular integrals in terms of so-called model operators conserving the same invariance structure. This paradigm applies to Zygmund-type operators and modulation invariant operators akin to the bilinear Hilbert transform, both of which are out of reach for dyadic-probabilistic methods. Concrete applications come from elliptic and dispersive PDE, operator theory and quasi-conformal mappings. The second related family of questions is motivated by pointwise convergence of bilinear ergodic averages for Banach-valued functions, a celebrated theorem by Bourgain in the scalar case. The approach is based on Banach-valued variational estimates for the truncated bilinear Hilbert transform. The central item in a further set of questions concerning directional singular integrals is a version of the Kakeya maximal estimate where tubular averages are replaced with averages over singular line segments, and more generally, n-dimensional subspaces. One source of motivation is the connection with Fourier restriction in higher codimensions. The methods involve algebra-geometric techniques such as polynomial partitioning on manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.