This Project Grant from the National Science Foundation Division of Mathematical Sciences provides $162,825 to support collaborative research in enumerative algebraic geometry at Virginia Polytechnic Institute and State University from August 1, 2022 to July 31, 2025. The research aims to resolve outstanding questions in enumerative algebraic geometry through investigations of intersection theory in moduli spaces of geometric figures including flag varieties, their cotangent bundles, bow varieties, and Hessenberg varieties. Techniques will include intersection theory, equivariant localization, symmetric functions, Hecke algebra actions, and geometric representation theory. This award falls under the NSF's Mathematical and Physical Sciences program (CFDA 47.049), which supports increasing scientific knowledge and understanding of major problems through advancement of the mathematical and physical sciences.