Geometric, Optimizational and Spectral Problems in Large Random Structures

大型随机结构中的几何、优化和谱问题

• 批准号: 1953848
• 负责人: Jian Ding
• 金额: $ 37.53万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953848&HistoricalAwards=false
• 关键词: Geometric; Optimizational; Spectral; Problems; Large

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.

该奖项支持的研究集中在大型随机结构上，这些结构具有许多身体学科的动力。 PI试图提高对数学物理和计算机科学中一系列基本模型和现象的理解，包括绝缘体 - 导电器过渡，计算能力在解决现实问题中的极限以及某些二维随机表面模型对共同场理论的重要性。此外，PI正在积极地在面向应用程序的项目上合作，而具体目标之一是帮助合并某些统计学习程序的理论基础。最后，PI打算为研究生和概率理论的博士后研究人员提供研究机会。PI将研究随机模型的限制行为，重点是几何，优化和频谱方面以及它们之间的相互作用。所考虑的随机过程涵盖了广泛的频谱，并且在各自的研究社区中至关重要。例如，高斯自由场的随机几何形状在二维概率上一直是一个惊人的研究主题，并且可以将密切相关的liouville量子重力表面视为随机二维Riemannian歧管的规范模型。随机约束满意度问题与旋转玻璃理论和复杂性理论有很强的联系，并且了解解决方案空间的相跃迁与算法过渡之间的联系是主要的兴趣。随机场模型是了解疾病的存在如何影响统计物理模型的行为的一个典型例子。安德森（Anderson）的随机定位是通过理解导体 - 压形器过渡的动机。为了在这些问题上取得进展，PI将试图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。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和广泛影响的评估来通过评估来支持的。