The theory of surfaces in geometry and relativity

This project utilizes powerful geometric concepts, known as classical surface theories, to unlock the mysteries of spacetime. This research will deepen our understanding of gravity, black holes, and the very fabric of spacetime. The project's findings can complement experimental data on phenomena like gravitational waves, currently heavily reliant on numerical simulations. Novel methods developed during this research might find applications in other areas of mathematics and physics. The research in this project will also promote interests in mathematics among undergraduate and graduate students and young researchers in the mathematical community. This project leverages the power of classical surface theories, including techniques like isometric embedding and the Gauss map, to investigate complex problems in differential geometry and general relativity. By focusing on the non-linear nature of spacetime, the research aims to:1.Evaluate the quasi-local mass of binary black holes: Develop a more precise method for calculating the combined mass and individual masses of these fascinating objects.2.Define angular momentum in general relativity: Establish a rigorous definition of angular momentum applicable to global solutions of Einstein's equations.3.Prove existence and regularity of a geometric flow: Mathematically demonstrate the existence and well-behaved nature of a specific type of geometric evolution. 4. Demonstrate duality in string theory: Reveal an underlying connection between two seemingly disparate equations within string theory, potentially leading to new avenues of exploration. These advancements promise to significantly contribute to our understanding of the universe and the power of mathematics in unraveling its mysteries. 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.