Topological recursion is a recent development in geometry that assigns an infinite family of quantities (called correlation functions) to a spectral curve. Although it plays an important role in many branches of mathematics, such as enumerative geometry, Gromov-Witten theory, and random matrix theory, it is still a poorly understood phenomenon. The goal of the proposed project is to establish a general mathematical framework underlying topological recursion. This structure is expressed through the newly introduced concepts of a topological recursion (TR) operad and TR co-operad, and allows one to understand the types of enumerative and geometric problems where Eynard-Orantin theory appears, a precise mechanism to calculate the spectral curve of such a problem, and, conversely, a concrete geometric or enumerative problem associated to an arbitrary spectral curve. To be specific, the combinatorial structure of Eynard-Orantin recursion is easily seen to be modeled on a TR operad, while in many specific examples where Eynard-Orantin recursion appears, such as intersection numbers of moduli spaces of stable curves, Hurwitz theory, Kontsevich's matrix integral, etc., there is a naturally appearing TR co-operad. In general, the Laplace transform of a TR operad is conjectured to determine the spectral curve. On the other hand, a variation of the Feynman transform (which appears in modular operads), takes a TR operad to a TR co-operad in the category of topological spaces, which comes equipped with a natural measure. The volume of the TR co-operad is conjectured to determine the Eynard-Orantin correlation functions which underlie the original TRoperad.
Successful completion of the project would go a long way to understanding the role and nature of topological recursion in general, and would provide several useful applications in enumerative geometry and Gromov-Witten theory. The area being investigated is the moduli space of Riemann surfaces, and related spaces. Such constructions arise in high energy physics and are crucial in our understanding of mirror symmetry, and more generally string theory. However, potential applications are not limited to theoretical physics. Because of the ubiquity of surfaces in everyday life, there are a surprising number of concrete applications for such an abstract subject. For example, ribbon graphs, one of the fundamental tools used to explore topological recursion, can be utilized in facial recognition algorithms. In addition, they appear in Feynman graph expansions, which are used to model particle collisions.
