Geometric, Optimizational and Spectral Problems in Large Random Structures

该奖项支持的研究重点是大型随机结构，这些结构具有来自多个物理学科的强烈动机。PI试图提高对数学物理和计算机科学中的一系列基本模型和现象的理解，包括绝缘体-导体转变、在解决实际问题时计算能力的限制，以及一些对保形场理论重要的二维随机表面模型。此外，统计研究所正在积极协作开展以应用为导向的项目，其具体目标之一是帮助巩固一些统计学习程序的理论基础。最后，PI旨在为概率论的研究生和博士后研究人员提供研究机会。PI将研究随机模型的极限行为，重点是几何、优化和谱方面以及它们之间的相互作用。所考虑的随机过程涉及的范围很广，在各自的研究界具有基本的重要性。例如，高斯自由场的随机几何一直是二维概率论中的一个现象级研究课题，与之密切相关的Liouville量子引力面可以被认为是随机二维黎曼流形的正则模型；随机约束满足问题与自旋玻璃理论和复杂性理论有很强的联系，理解解空间的相变和算法跃迁之间的联系是人们感兴趣的；随机场模型是理解无序存在如何影响统计物理模型行为的经典例子；随机薛定谔算符的Anderson局部化是通过理解导体-绝缘体相变来实现的。为了在这些问题上取得进展，PI将尝试从物理学中引入新的见解，开发新的数学工具，并确定不同数学分支之间的新联系。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The research supported by this award focuses on large random structures that have strong motivation from a number of physical disciplines. The PI attempts to improve understanding on a range of fundamental models and phenomena in mathematical physics and computer science, including the insulator-conductor transition, the limit of computing power in solving realistic problems, and some two-dimensional random surface models of importance to conformal field theory. In addition, the PI is actively collaborating on application-oriented projects, and one of the specific goals is to help consolidating theoretical foundations for some statistical learning procedures. Finally, the PI intends to provide research opportunities for graduate students and postdoctoral researchers in probability theory.The PI will study limiting behavior for stochastic models with emphasis on geometric, optimizational and spectral aspects as well as interactions among them. The stochastic processes under consideration cover a wide range of spectrum and are of fundamental importance in respective research communities. For instance, random geometry of Gaussian free field has been a phenomenal research topic in two-dimensional probability, and the closely related Liouville quantum gravity surfaces can be thought of as the canonical models of random two-dimensional Riemannian manifolds; random constraint satisfaction problems have strong connections with spin glass theory and complexity theory, and it is of major interest to understand the connections between the phase transitions of the solution spaces and the algorithmic transitions; random field model has been a classic example in understanding how presence of disorder affects the behavior of statistical physics models; Anderson localization for random Schroedinger operators is motivated by understanding the conductor-insulator transition. In order to make progress on these problems, the PI will attempt to bring in new insights from physics, develop new mathematical tools and identify new connections among different mathematical branches.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.