Randomized affine isoperimetry and concentration phenomena

拟议的研究涵盖了概率、凸几何和分析等核心主题。凸几何是研究等周原理的基础，等周原理是支配形状和大小之间基本关系的定律。最著名的例子是经典的等周不等长，它断言在给定周长的所有形状中，圆包围的面积最大。这些原理构成了大量极端问题的基础，例如，在数学物理、信息论、最优化等领域。PI和他的合著者已经证明，全局几何特征可以是局部随机结构的结果；他的概率工具揭示了比标准原理更明显的新的和更多的定量信息。相反，底层形状的几何形状可能会对概率产生影响。例如，这是通过用更广泛的依赖结构取代独立条件，从而使概率结果更广泛地适用而产生的。这项研究的一个主要目标是使用几何考虑因素，如许多对称性的存在，作为取代独立性的指导。在一个密切相关的方向上，在应用科学中，“维度诅咒”指的是这样一个概念，即增加一个系统的维度会伴随着复杂程度的笨拙增加。另一方面，现代概率和凸几何的一个显著特征是，增加维度会带来意想不到的好处：模式必须简单地凭借高维而产生。在数学上，这被称为“测量集中现象”，它是有效处理大数据集、压缩信号、降低算法复杂性等方面的基础。PI和co-PI将开发新的工具，在精确的渐近标度上提供相当准确的信息，这在应用中尤其需要。这里，如上所述，等周原理指导着该理论的发展。PI和共同PI将教授关于这些主题的研究生课程，这将成为吸引学生参与当前研究的极好场所。这类课程可能会让计算机科学、统计学或工程专业的学生感兴趣，这些学生的研究主要依赖于数学。该项目的重点是高维概率律的集中性质，特别是边际律，因为它们与小偏差不等式和非渐近随机矩阵理论有关。利用仿射等周原理，PI将研究边缘分布良好有界的准则，特别是在仿射不变性质等非独立正则性假设下。他们还将研究高维欧氏空间上范数的浓度性质，建立在一些经典赋范空间的Milman随机版本的Dvoretzky定理的co-Pi的精化基础上，通过使用超浓缩技术和其他精化工具来绕过通过Lipschitz常数的标准方法。PI和co-PI将泛函集中的研究推广到欧氏空间中线性子空间的Grassman流形的多维环境中。这是上述各种问题的自然统一设置：概率分布的边缘、凸体的渐近理论、随机几何和随机化的等周不等式。因此，更好地理解格拉斯曼随机性将具有广泛的应用前景。

The proposed research covers core topics in probability, convex geometry and analysis. Convex geometry is the bedrock for studying isoperimetric principles, laws that govern fundamental relationships between shapes and their size. The most famous such example is the classical isoperimetric inequality asserting that among all shapes of a given perimeter, circles enclose the largest area. Such principles underlie a wealth of extremal problems, e.g., in mathematical physics, information theory, optimization, among others. The PI and his coauthors have shown that global geometric features can be consequences of local random structure; his probabilistic tools reveal new and more quantitative information than is apparent from the standard principles. Conversely, the geometry of the underlying shapes can have implications in probability. This arises, for example, by replacing independence conditions by broader dependence structures, thereby making probabilistic results more broadly applicable. A major goal of the research is to use geometric considerations such as the presence of many symmetries as a guide for the replacement of independence. In a closely related direction, in applied sciences the "curse of dimensionality" refers to the notion that increasing a system's dimension comes with an unwieldy increase in complexity. On the other hand, a distinguishing feature of modern probability and convex geometry is that increasing the dimension brings unexpected benefits: patterns must arise simply by virtue of high-dimensionality. Mathematically, this is referred to as the "concentration of measure phenomenon" and it is fundamental in dealing effectively with large data sets, compression of signals, reducing complexity of algorithms, to name a few. The PI and co-PI will develop new tools to give considerably more accurate information on refined asymptotic scales, which are especially needed for applications. Here, as above, isoperimetric principles guide the development of the theory. The PI and co-PI will teach graduate courses on these topics, which will serve as excellent venues for engaging students in current research. Such courses may be of interest to students in computer science, statistics, or engineering whose research depends vitally on mathematics.The project centers on concentration properties of high-dimensional probability laws, particularly for marginal laws due to their connection to small deviation inequalities and non-asymptotic random matrix theory. Using affine isoperimetric principles, the PIs will investigate criteria for well-boundedness of marginal distributions, especially under non-independence regularity assumptions such as affine invariance properties. They will also study concentration properties of norms on high-dimensional Euclidean spaces, building on the co-PI's refinements of Milman's random version of Dvoretzky's theorem for some classical normed spaces, circumvent the standard approach via Lipschitz constants by using super-concentration techniques and other refined tools. The PI and co-PI will extend the study of concentration of functionals to the multi-dimensional setting of Grassmannian manifold of linear subspaces of Euclidean space. This is a natural unified setting for the various problems above: marginals of probability distributions, the asymptotic theory of convex bodies, stochastic geometry and randomized isoperimetric inequalities. Consequently, a better understanding of the associated randomness on the Grassmannian will have diverse applications.