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.

该奖项支持的研究重点是大型随机结构，这些结构受到许多物理学科的强烈推动。 PI 试图增进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-导体转变、解决实际问题的计算能力极限，以及对共形场论重要的一些二维随机表面模型。此外，PI 正在积极合作开展面向应用的项目，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 打算为概率论方面的研究生和博士后研究人员提供研究机会。PI 将研究随机模型的限制行为，重点是几何、优化和谱方面以及它们之间的相互作用。正在考虑的随机过程涵盖了广泛的范围，并且在各自的研究界中具有根本重要性。例如，高斯自由场的随机几何一直是二维概率领域的一个现象级研究课题，密切相关的刘维尔量子引力面可以被认为是随机二维黎曼流形的规范模型；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，了解解空间的相变与算法转换之间的联系非常有意义；随机场模型是理解无序的存在如何影响统计物理模型行为的经典例子；随机薛定谔算子的安德森定位是通过理解导体-绝缘体转变来激发的。为了在这些问题上取得进展，PI将尝试引入物理学的新见解，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的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。