This project supports US-based mathematicians, postdoctoral researchers, and graduate students to participate in an international research event at the Okinawa Institute of Science and Technology (OIST), Okinawa, Japan, August 31–October 30, 2026. The goal of the two-month program is to bring together mathematicians and physicists to investigate fundamental questions about the structure of space, geometric form, and the physical laws governing our universe. The venue will create a sustained and concentrated research environment where insights from one field can inform and accelerate progress in the others. At its core, the program focuses on gauge theory, a theoretical construction originally developed in physics to describe elementary particles and the fundamental forces behind their interactions. Gauge theory has since become a powerful tool in modern mathematics. Using gauge-theoretic ideas, mathematicians can study the properties of curved spaces and uncover deep connections between seemingly disparate areas of mathematics, often suggested by considerations in theoretical physics. In terms of broader impacts, the program places strong emphasis on training and outreach. NSF support will prioritize early-career researchers, including graduate students and postdoctoral fellows, providing them with valuable opportunities for international collaboration and professional development. The program will also feature an autumn school, specialized workshops, and mentoring activit