CAREER: Mean Field Spin Glasses and Related Applications

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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.