Renormalization Group Flows, Embedding Theorems, and Applications

数学的一个中心主题是理解空间的本质。数学家使用流形这个术语来形式化弯曲空间的概念，弯曲空间是由更熟悉的欧几里得空间拼凑而成的。这个想法可以追溯到19世纪中期黎曼的工作，然而关于流形的几个问题仍然知之甚少。在20世纪50年代，纳什证明了惊人的嵌入定理，确定了考虑流形的两种不同方式，内在流形和外在流形。例如，我们可以想象，在地球上爬行的蚂蚁测量地球上的长度（内在的），在月球上的观察者测量地球上的长度（外在的）。这个项目的主要目的是从现代的角度重新审视纳什的嵌入定理，扩大它们在科学和技术问题中的应用范围。本项目的方法可应用于物理中无序系统的描述（如湍流流体流动）和人工智能算法的设计（如流形学习）。该项目通过指导研究生进行包括严格数学和数值计算在内的博士项目，促进了科学劳动力的发展。该项目的一个特定目标是可视化嵌入的流形，以便将纳什的愿景传达给广泛的科学观众和公众。本课题的主要技术目标是开发一种新的方法来构造非线性偏微分方程的吉布斯测度。该方法依赖于将PDE提升到高斯测度空间，并设计新的热流，称为重整化群流，通过在越来越精细的尺度上通过带限制近似将PDE的子解改进为解。将采用数值方法和分析方法。分析方法源于纳什的技术，但其概念基础是对长度测量的贝叶斯描述，而不是底层空间的性质。这种重点的转移提供了一个框架，将黎曼流形的嵌入问题与度量空间和图的嵌入问题统一起来。它通过统计物理中临界指数的研究来识别嵌入的临界指数问题。与这些解析方法相对应的数值方法是求解非线性偏微分方程的一类新的随机内点法。本文的主要目标是探索PDE可以通过松弛、半定规划和马尔可夫链蒙特卡罗方法等优化技术来解决，而不是基于空间离散化的传统方法。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。

A central theme in mathematics is to understand the nature of space. Mathematicians use the term manifold to formalize the notion of a curved space built by piecing together patches of more familiar Euclidean space. This idea dates to the work of Riemann in the mid-1800s, yet several questions about manifolds remain poorly understood. In the 1950s, Nash proved striking embedding theorems identifying two distinct ways of thinking about manifolds, intrinsic and extrinsic. One can imagine for example, the measurement of lengths on earth by an ant crawling around (intrinsic) and the measurement of lengths on earth by an observer on the moon (extrinsic). The main purpose of this project is to revisit Nash's embedding theorems from a modern perspective, expanding their range of applications to problems in science and technology. Approaches of this project have applications to the description of disordered systems in physics (such as turbulent fluid flows) and the design of algorithms for artificial intelligence (such as manifold learning). The project contributes to the development of the scientific workforce through the mentoring of graduate students in PhD projects that include both rigorous mathematics and numerical computations. A particular goal of the project is to visualize the embedded manifolds in order to communicate Nash’s vision to a broad scientific audience and the public at large.The main technical goal of this project is to develop a new method to construct Gibbs measures for nonlinear partial differential equations. The method relies on lifting the PDE into a space of Gaussian measures and designing new heat flows, termed renormalization group flows, that improve subsolutions of the PDE into solutions through band-limited approximations at increasingly fine scales. Both numerical and analytical approaches will be pursued. The analytical method has its roots in Nash’s techniques, but its conceptual foundation is a Bayesian description of the measurement of length, not the nature of the underlying space. This shift in emphasis provides a framework that unifies the embedding problem for Riemannian manifolds with that of metric spaces and graphs. It identifies the problem of critical exponents for embeddings with the study of critical exponents in statistical physics. The numerical counterpart to these analytical methods is a new class of stochastic interior point methods for the solution of nonlinear PDE. The primary goal here is to explore the idea that PDE may be solved by optimization techniques, such as relaxation, semidefinite programming, and Markov chain Monte Carlo methods, rather than the traditional methods based on spatial discretizations.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.