PRIMES: Foundations, Analysis, Implementation, and Applications of Nonlocal Vector Calculus to Mathematics of Intelligences

This project aims to explore the intersections of nonlocal vector calculus and the mathematics of intelligences, which are foundational to understanding complex systems and intelligent phenomena in nature and technology. By advancing mathematical theories and models, this research will contribute to fields ranging from artificial intelligence to natural and cognitive sciences.

The project's outcomes will provide new tools and insights that can enhance technological innovations and scientific discoveries. Intentionally involving graduate and undergraduate students from California State University Channel Islands (CSUCI), a Hispanic Serving Institution, in this interdisciplinary research taking place at the Institute for Applied Mathematics (IPAM) will also help train the next generation of mathematicians and scientists, fostering community, diversity, and inclusion in these fields.

Through this collaboration with IPAM, the project seeks to elevate the research capacity at CSUCI, contribute to the broader understanding of intelligences, and actively nurture a diverse and vibrant community of scholars in the mathematical sciences.

The research objectives center on conducting focused investigations at the intersections of nonlocal analysis, dispersive PDEs, and the mathematics of intelligences. The primary goals are to formulate new theoretical models, prove significant properties, and apply these models to problems in both artificial and natural intelligent systems. The strength of introducing a nonlocal framework, particularly through the use of nonlocal operators and singular integral operators, lies in its ability to model discontinuous phenomena, as has been done successfully in the contexts of materials damage, fracture, swarms, and, most recently, neural networks.

Exploration topics include spatial interactions, long-range dependencies, nonlocal uncertainty, wave propagation, and information transfer, all as they relate to the mathematics of intelligences. The PI will undertake a one-year research fellowship and attend the program on Mathematics of Intelligences at IPAM during Fall 2024, providing a platform for contributions to the mathematics of intelligences through knowledge dissemination.

Research collaborations will be established and nurtured to create lasting partnerships with collective expertise.

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.