CAREER: Mean Field Spin Glasses and Related Applications

在20世纪70年代，理论物理学家发明了一些统计力学模型，称为自旋玻璃，以研究某些合金的奇怪行为，如CuMn。在数学中，它们是纯粹的概率对象，拥有极其复杂的依赖结构，并表现出几个关键特征，这些特征在各种科学领域产生的许多随机组合系统中普遍存在。因此，自旋玻璃模型经常被引用作为复杂系统的例子，相应的研究在理解计算机科学、数据科学等领域出现的许多现实问题方面有很大的用处。该项目旨在提高目前对平均场自旋玻璃模型的数学理解，并为数据科学中的相关应用建立理论基础。该项目的一个组成部分是教育活动的发展，包括指导研究生和博士后研究员，开发高级课程，并监督本科生暑期研究。该项目涉及研究功能序参量的结构及其与平均场自旋玻璃的涨落和能量景观的联系。此外，PI计划通过基于Thouless-Anderson-Palmer方法的经典统计力学研究平均场自旋玻璃模型，并分析其与Parisi理论的关系。PI还将研究具有更复杂依赖结构的自旋玻璃模型，包括二分模型和Levy自旋玻璃。此外，PI打算在自旋玻璃理论的方法中研究数据科学中产生的一些随机组合优化问题，如检测问题和半正定规划。基于这些研究项目，教育部分包括组织概率暑期课程和指导研究生，本科生以及博士后研究员。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。

In the 1970s theoretical physicists invented a number of statistical mechanical models, called spin glasses, in order to study the strange behavior of certain alloys, such as CuMn. In mathematics, they are purely probabilistic objects possessing extremely intricate dependence structures and exhibit several crucial features that are commonly shared in many randomized combinatorial systems arising from various scientific fields. As a result, spin glass models are very often cited as examples of complex systems and the corresponding research has been of great use in understanding many real-world problems emerging from, such as, computer science, data science, and many others. This project aims to improve the current mathematical understanding of the mean field spin glass models and to establish theoretical foundations for related applications in data science. An integral part of the project is the development of educational activities including mentoring graduate students and postdoc fellows, developing advanced courses, and supervising undergraduate summer research. The project involves the study of the structure of the functional order parameter and its connection to the fluctuation and the energy landscape of the mean field spin glasses. In addition, the PI plans to investigate the mean field spin glass models by means of the classical statistical mechanics based on the Thouless-Anderson-Palmer approach and analyze its relation to the Parisi theory. The PI will also study spin glass models with more complicated dependence structures including the bipartite model and the Levy spin glass. Furthermore, the PI intends to study some randomized combinatorial optimization problems arising from data science in the approach of spin glass theory such as the detection problems and positive semi-definite programming. Based on these research projects, the educational component involves organizing probability summer programs and mentoring graduate, undergraduate students as well as postdoc fellows.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.