The aim of this project is to extend the theory of calculus to more complex geometric settings, particularly those relevant to theoretical physics. Traditionally, calculus is defined on flat spaces like lines or planes, while modern differential geometry expands these concepts to smoothly curved spaces—such as spheres or donuts—and their higher-dimensional analogues, called manifolds. This theory plays a central role in physics, from Einstein’s description of gravity as the curvature of spacetime to the Standard Model of particle physics. The algebraic process of solving equations corresponds geometrically to the intersection of graphs, a principle that extends naturally to manifolds. However, intersections of manifolds are not always manifolds themselves, rendering differential geometry insufficient. The PI has made significant contributions to derived differential geometry (DDG), an advanced framework designed to handle such non-smooth intersections. Yet, integration—a cornerstone of calculus—has not been fully developed in this setting. The first aim of the project is to fill in this gap by building a robust theory of integration in DDG, with particular relevance to the computation of Feynman path integrals in physics. The second aim is to generalize geometric quantization—a powerful method traditionally used to describe the transition from classical mechanics to quantum mechanics—to more sophisticated systems such as classical field theories. Classical mechanics describes