Research project This project explores the best methods for introducing tensors and geometric metrics to the theory.
Recently, Hofer-Wysocki-Zehnder developed a model allowing for a smooth and local change of dimensions, a revolutionary development. However, this theory is still in its developmental stage. This project explores the best methods for introducing tensors and geometric metrics to the theory.
Manifold theory plays a crucial role in modern models like information geometry, Hamiltonian mechanics, digital signal processing, and Einstein's theory of relativity. However, manifolds have a significant limitation: they must have the same dimension locally. For example, a ball with a wire soldered cannot be modeled as a manifold. Recently, Hofer-Wysocki-Zehnder developed a model allowing for a smooth and local change of dimensions, a revolutionary development. They used this model to solve compactness problems in symplectic geometry. This model, which combines a generalized differential geometry and a generalization of classical nonlinear Fredholm theory, has significant potential. However, this theory is still in its developmental stage. This project explores the best methods for introducing tensors and geometric metrics to the theory. It is worth noting that there currently needs to be tensors and metrics in this theory.
