This Project Grant award from the National Science Foundation's (NSF) Mathematical and Physical Sciences (CFDA 47.049) program provides $300,000 to Brown University to develop a new mathematical framework for the Nash embedding theorems. The project aims to align the fundamental mathematics with cutting-edge scientific applications, especially in artificial intelligence. Key products will include rigorous analyses of Riemannian Langevin equations, new constructions of Brownian motion on Riemannian manifolds, and optimization and sampling algorithms that leverage the geometric properties of deep neural networks. The award also supports the training of PhD students in STEM fields through mentoring. The work will contribute to advancing scientific understanding in areas like fluid turbulence and the representation of big data using deep learning. The award commenced on August 15, 2024 and is scheduled for completion on July 31, 2027.
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